What does the derivative of a function at a point describe?

[Frank Castle]

I understand that the derivative of a function ##f## at a point ##x=x_{0}## is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where ##\Delta x## is a small change in the argument ##x## as we "move" from ##x=x_{0}## to a neighbouring point ##x=x_{0}+\Delta x##.
What confuses me is how to interpret its meaning correctly, that is, what does the derivative ##f'(x_{0})## actually describe?

On Wikipedia it says that "the derivative of a function ##f(x)## of a variable ##x## is a measure of the rate at which the value of the function changes with respect to the change of the variable". However, the function has a particular constant value, ##f(x_{0})## at a given point ##x=x_{0}## so how can one meaningfully discuss the rate at which the value of the function is changing at that point?

Would it be correct to interpret the derivative of a function at a point as quantifying the slope of the line tangent to the curve (corresponding to the coordinate plot of the function ##f(x)##) at that point? Is the whole idea that when taking the derivative of a function one isn't ever considering the value of a function at a given point, one considers the function over a given neighbourhood. Upon taking the derivative one subsequently evaluates it at a given point and this quantifies the instantaneous rate of change in the value of the function (with respect to a change in its input variable ##x##) at that point. For a linear function, this rate of change is constant, i.e. the rate at which the value of the function is changing (with respect to a change in its input variable ##x##) at each point is the same. For a more general (continuous) function, the derivative will itself be a function and its value will change from point to point, however, at each given point its value at that point will still equal the slope of the line (corresponding to a the coordinate plot of a linear function) tangent to the curve (corresponding to the coordinate plot of the more general function we're considering) at that point?!

Apologies for such a basic question, but I've really got a mental block in my head about this and want to clear it up.

 

[Samy_A]

Would it be correct to interpret the derivative of a function at a point as quantifying the slope of the line tangent to the curve (corresponding to the coordinate plot of the function ##f(x)##) at that point?

Yes.

 

[Mark44]

I understand that the derivative of a function ##f## at a point ##x=x_{0}## is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where ##\Delta x## is a small change in the argument ##x## as we "move" from ##x=x_{0}## to a neighbouring point ##x=x_{0}+\Delta x##.
What confuses me is how to interpret its meaning correctly, that is, what does the derivative ##f'(x_{0})## actually describe?

On Wikipedia it says that "the derivative of a function ##f(x)## of a variable ##x## is a measure of the rate at which the value of the function changes with respect to the change of the variable". However, the function has a particular constant value, ##f(x_{0})## at a given point ##x=x_{0}## so how can one meaningfully discuss the rate at which the value of the function is changing at that point?

The function has a particular value at a specified point, but the function values are not constant, in general. A different explanation of the meaning of the derivative of a function at a point is that ##f'(x_0)## is the slope of the tangent line at the point ##(x_0, f(x_0))##.

Would it be correct to interpret the derivative of a function at a point as quantifying the slope of the line tangent to the curve (corresponding to the coordinate plot of the function ##f(x)##) at that point? Is the whole idea that when taking the derivative of a function one isn't ever considering the value of a function at a given point, one considers the function over a given neighbourhood. Upon taking the derivative one subsequently evaluates it at a given point and this quantifies the instantaneous rate of change in the value of the function (with respect to a change in its input variable ##x##) at that point. For a linear function, this rate of change is constant, i.e. the rate at which the value of the function is changing (with respect to a change in its input variable ##x##) at each point is the same. For a more general (continuous) function, the derivative will itself be a function and its value will change from point to point, however, at each given point its value at that point will still equal the slope of the line (corresponding to a the coordinate plot of a linear function) tangent to the curve (corresponding to the coordinate plot of the more general function we're considering) at that point?!

Apologies for such a basic question, but I've really got a mental block in my head about this and want to clear it up.

 

[PeroK]

Apologies for such a basic question, but I've really got a mental block in my head about this and want to clear it up.

It's quite a common question and I think a lot of people have to digest this idea at some stage. You have, in fact, given a fairly good description of what a derivative is.

 

[Frank Castle]

You have, in fact, given a fairly good description of what a derivative is.

Thanks. How could I improve it though? Are there any deeper insights that I'm missing?

The function has a particular value at a specified point, but the function values are not constant, in general.

By this do you mean that the function has a different value at each point? I thought I understood the meaning of the derivative of a function, but the description on Wikipedia (that I quoted in my post) has thrown a spanner in the works. I'm unsure how to interpret it correctly?!

 

[PeroK]

Thanks. How could I improve it though? Are there any deeper insights that I'm missing?

In terms of a simple derivative there's perhaps not much more to say. In mathematical terms, though, the derivative is what it is formally defined to be. None of the descriptions actually matter one iota.

The next step is understanding things like the chain rule, the relationship between differentiation and integration (Fundamental Theorem of Calculus) and partial derivatives for a function of several variables; or, of course, functions of a complex variable.

What level are you studying at?

 

[Frank Castle]

What level are you studying at?

Undergraduate level (I'm ashamed to say). I understand how to use the derivative and its associated rules, etc. and I thought I understood intuitively what a derivative is, but now I'm not so sure, the description I read on Wikipedia has really thrown a spanner in the works for me (I'm not happy with simply accepting definitions when I learn maths, I want to have an intuitive idea of what the particular operation is describing). What the sticking point for me is, how do I make sense of the statement:

"the derivative of a function f(x)f(x) of a variable xx is a measure of the rate at which the value of the function changes with respect to the change of the variable"

What exactly does this statement mean? Is it simply that although the function has a fixed value at that point the value of the function will in general be changing from point to point and the derivative evaluated at a particular point describes the rate at which its value is changing at that point in the sense that if a move away from that point (say ##x_{0}##) by a small amount ##\Delta x##, the value of the function ##f## will approximately change by an amount ##f'(x_{0})\Delta x##?

 

[PeroK]

Undergraduate level (I'm ashamed to say). I understand how to use the derivative and its associated rules, etc. and I thought I understood intuitively what a derivative is, but now I'm not so sure, the description I read on Wikipedia has really thrown a spanner in the works for me (I'm not happy with simply accepting definitions when I learn maths, I want to have an intuitive idea of what the particular operation is describing). What the sticking point for me is, how do I make sense of the statement:

There are a hundred ways to describe something. Personally, I wouldn't worry if you don't understand a particular person's way of describing something, as long as you understand someone else's. For example, you said "the derivative is itself a function", which I think is better than the extract from Wiki. From a maths point of view that is crucial. Never forget that, by the way: the derivative is itself a function.

 

[Mark44]

The function has a particular value at a specified point, but the function values are not constant, in general.

By this do you mean that the function has a different value at each point?

I'm not sure why you are puzzled by this. Consider the graph of ##y = f(x) = x^2##. The graph is a parabola with its vertex at the origin. The derivative is f'(x) = 2x. At the point (1, 1) on the parabola, f'(1) = 2, so the slope of the tangent line is 2 at this point on the parabola. At the vertex of the parabola, f'(0) = 0, so the tangent line at (0, 0) is horizontal (its slope is 0).

 

[Frank Castle]

I'm not sure why you are puzzled by this. Consider the graph of y=f(x)=x2y = f(x) = x^2. The graph is a parabola with its vertex at the origin. The derivative is f'(x) = 2x. At the point (1, 1) on the parabola, f'(1) = 2, so the slope of the tangent line is 2 at this point on the parabola. At the vertex of the parabola, f'(0) = 0, so the tangent line at (0, 0) is horizontal (its slope is 0).

I think what confuses me is how one can consider the rate of change in a value at a single point, surely you can only discuss it relative to another point? It would make sense to me that the derivative of a function at a point describes the rate at which its value changes as you move away from that point. Before, I always thought of the derivative at a particular point as quantifying the slope of the line tangent to the function curve at that point...
In the linear case it is easy for me to understand as the derivative is simply the slope (gradient) and is constant ##m##, thus the function value ##y=f(x)=mx+b## is changing at a rate of ##m## units of ##y## per unit of ##x## at each point along the line. Can one simply generalise this to an arbitrary continuous function, except that now the derivative is not constant and so will correspond to a different slope (of a different line tangent to the curve) at each point. However, at a given point ##x_{0}##, the derivative ##f'(x_{0})## equals the slope of the line that is tangent to the curve ##y=f(x)## at the point ##x_{0}## and hence the value of the function at that point is changing at a rate of ##f'(x_{0})## units of ##y## per unit of ##x##?!

 

[Mark44]

I think what confuses me is how one can consider the rate of change in a value at a single point, surely you can only discuss it relative to another point?

Another way to think about it is this: Imagine that you're a bug traveling along the curve, from left to right. At the point in question, which direction would you be facing?

Do you know what a secant line is? If not, it's a line that connects two points on a curve. The slope of the secant line between ##(x_0, f(x_0))## and ##(x_0 + h, f(x_0 + h))## has a slope of ##\frac{f(x_0 + h) - f(x_0)}{x_0 + h - x_0}##, which is change in y values divided by change in x values, or rise over run. As you take smaller and smaller values of h you get secant lines that are closer in slope to the tangent line at ##(x_0, f(x_0))##. The derivative of f at ##x_0## is defined by this limit:
$$f'(x_0) = \lim_{h \to 0}\frac{f(x_0 + h) - f(x_0)}{h}$$

It would make sense to me that the derivative of a function at a point describes the rate at which its value changes as you move away from that point? Before I always thought of the derivative at a particular point as quantifying the slope of the line tangent to the function curve at that point...

Yes, that's what it is (and what I already said).

 

[Frank Castle]

Yes, that's what it is (and what I already said).

Thanks for the info. Sorry to repeat what you'd already said.
Also, would the description that I added to the end of my last post be correct? (i.e.

In the linear case it is easy for me to understand as the derivative is simply the slope (gradient) and is constant mm, thus the function value y=f(x)=mx+by=f(x)=mx+b is changing at a rate of mm units of yy per unit of xx at each point along the line. Can one simply generalise this to an arbitrary continuous function, except that now the derivative is not constant and so will correspond to a different slope (of a different line tangent to the curve) at each point. However, at a given point x0x_{0}, the derivative f′(x0)f'(x_{0}) equals the slope of the line that is tangent to the curve y=f(x)y=f(x) at the point x0x_{0} and hence the value of the function at that point is changing at a rate of f′(x0)f'(x_{0}) units of yy per unit of xx?!

)

 

[Mark44]

Thanks for the info. Sorry to repeat what you'd already said.
Also, would the description that I added to the end of my last post be correct? (i.e.
)In the linear case it is easy for me to understand as the derivative is simply the slope (gradient) and is constant ##m##, thus the function value ##y=f(x)=mx+b## is changing at a rate of ##m## units of ##y## per unit of ##x## at each point along the line. Can one simply generalise this to an arbitrary continuous function, except that now the derivative is not constant and so will correspond to a different slope (of a different line tangent to the curve) at each point. However, at a given point ##x_{0}##, the derivative ##f'(x_{0})## equals the slope of the line that is tangent to the curve ##y=f(x)## at the point ##x_{0}## and hence the value of the function at that point is changing at a rate of ##f'(x_{0})## units of ##y## per unit of ##x##?!

Yes

 

[HallsofIvy]

However, the function has a particular constant value, f(x0)'>f(x0)f(x0) f(x_{0}) at a given point x=x0'>x=x0x=x0 x=x_{0} so how can one meaningfully discuss the rate at which the value of the function is changing at that point?

That is a crucial question and is, largely, the reason "Calculus" was created. Both Newton and Leibniz were concerned with the problem of finding how gravity determined the orbits of planets. They knew, of course, that "force equals mass times acceleration". But that caused a conceptual problem! Since it was clear to both that gravitational force depended on the distance from the sun. At a specific instant, the distance from the sun to a planet is a fixed number (not "constant"- that should only be applied to functions, not individual numerical values) while "speed" is "change in distance divided by change in time" while "acceleration" is "change in speed divided by change in time". They require "change in time" so can't be at a given instant. So how can acceleration, which requires a change in time, depend on distance, which does not?

The solution to that problem was to use a "limiting process" so that we can define "instantaneous" speed and acceleration.

 

[Frank Castle]

That is a crucial question and is, largely, the reason "Calculus" was created. Both Newton and Leibniz were concerned with the problem of finding how gravity determined the orbits of planets. They knew, of course, that "force equals mass times acceleration". But that caused a conceptual problem! Since it was clear to both that gravitational force depended on the distance from the sun. At a specific instant, the distance from the sun to a planet is a fixed number (not "constant"- that should only be applied to functions, not individual numerical values) while "speed" is "change in distance divided by change in time" while "acceleration" is "change in speed divided by change in time". They require "change in time" so can't be at a given instant. So how can acceleration, which requires a change in time, depend on distance, which does not?

The solution to that problem was to use a "limiting process" so that we can define "instantaneous" speed and acceleration.

So is the idea that by defining the derivative in terms of a limit we can consistently quantify the rate of change in the function at a particular point? Is the limit definition basically stating that we can make ##\Delta x## arbitrarily close to zero, without actually equaling zero, such that within this arbitrarily small interval ##(x_{0},x_{0}+\Delta x)##, the ratio ##\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}## is equal to the value ##f'(x_{0})##, i.e. the slope of the tangent line to the point ##x_{0}##, and quantifies the rate at which the value of ##f## is changing due to a change in ##x## at the particular point ##x_{0}##?!

If the value of ##\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}## as ##\Delta x= x-x_{0}## approaches 0 is independent of the manner in which ##\Delta x## approaches 0, i.e. independent of whether ##\Delta x## approaches 0 from the left hand side of ##x_{0}## or the right hand side of ##x_{0}##, then it's limiting value exists and we say that is exactly equal to the rate of change in ##f(x)## relative to ##x## at the point ##x_{0}##.

Would this be correct at all?

 

[Mark44]

So is the idea that by defining the derivative in terms of a limit we can consistently quantify the rate of change in the function at a particular point?

Is the limit definition basically stating that we can make ##\Delta x## arbitrarily close to zero, without actually equaling zero, such that within this arbitrarily small interval ##(x_{0},x_{0}+\Delta x)##, the ratio ##\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}## is equal to the value ##f'(x_{0})##

No, not equal. For a specific nonzero value of ##\Delta x##, the ratio ##\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}## gives the slope of the secant line, the line that joins ##(x_0, f(x_0))## and ##(x_0 + \Delta x, f(x_0 + \Delta x))##. The smaller that ##\Delta x## is, the closer the slope of the secant line will be to the slope of the tangent line at ##(x_0, f(x_0))##; i.e., ##f'(x_0)##. But as long as ##\Delta x## is different from zero, the slope of the secant line will be different from ##f'(x_0)##.

, i.e. the slope of the tangent line to the point ##x_{0}##, and quantifies the rate at which the value of ##f## is changing due to a change in ##x## at the particular point ##x_{0}##?!

If the value of ##\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}## as ##\Delta x= x-x_{0}## approaches 0 is independent of the manner in which ##\Delta x## approaches 0, i.e. independent of whether ##\Delta x## approaches 0 from the left hand side of ##x_{0}## or the right hand side of ##x_{0}##, then it's limiting value exists and we say that is exactly equal to the rate of change in ##f(x)## relative to ##x## at the point ##x_{0}##.

Would this be correct at all?

 

[Frank Castle]

No, not equal. For a specific nonzero value of ΔxΔx\Delta x, the ratio f(x0+Δx)−f(x0)Δxf(x0+Δx)−f(x0)Δx\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x} gives the slope of the secant line, the line that joins (x0,f(x0))(x0,f(x0))(x_0, f(x_0)) and (x0+Δx,f(x0+Δx))(x0+Δx,f(x0+Δx))(x_0 + \Delta x, f(x_0 + \Delta x)). The smaller that ΔxΔx\Delta x is, the closer the slope of the secant line will be to the slope of the tangent line at (x0,f(x0))(x0,f(x0))(x_0, f(x_0)); i.e., f′(x0)f′(x0)f'(x_0). But as long as ΔxΔx\Delta x is different from zero, the slope of the secant line will be different from f′(x0)f′(x0)f'(x_0).

But I thought that the whole point of a limit was that the input variable is never actually equal to the value it is approaching? (At least that's what I've read in a couple of introductions to limits)

 

[Mark44]

But I thought that the whole point of a limit was that the input variable is never actually equal to the value it is approaching? (At least that's what I've read in a couple of introductions to limits)

Right, and I didn't say that it was.

My objection was to what you wrote:

Is the limit definition basically stating that we can make ##\Delta x## arbitrarily close to zero, without actually equaling zero, such that within this arbitrarily small interval ##(x_{0},x_{0}+\Delta x)##, the ratio ##\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}## is equal to the value ##f'(x_{0})##

##f'(x_0)## is NOT equal to the ratio above at each point in some interval ##[x_0, x_0 + \Delta x]##.

It might be helpful to actually work with numbers and a specific function. Let y = f(x) = x² and find the derivative of this function at the point (1, 1), using the difference quotient with several values for ##\Delta x##.

##\Delta x## = 0.1
Ratio - ##\frac{f(1.1) - f(1)}{.1} = \frac{1.21 - 1}{.1} = 2.1##

##\Delta x## = 0.01
Ratio - ##\frac{f(1.01) - f(1)}{.01} = \frac{1.0201 - 1}{.01} = 2.01##

Continue in this fashion for smaller values of ##\Delta x##. You should see that the difference quotient (identified above as the ratio, and which gives the slope of the secant line) will be slightly larger than the value of f'(1), which is 2. However, smaller values of ##\Delta x## give values of the difference quotient that are closer to, but not equal to the value of f'(1).

 

[Frank Castle]

f′(x0)f′(x0)f'(x_0) is NOT equal to the ratio above at each point in some interval [x0,x0+Δx][x0,x0+Δx][x_0, x_0 + \Delta x].

Sorry, very bad (and incorrect) phrasing on my part. I was trying to reword the definition of the limit, specifically the part ##0<\lvert\Delta x\rvert <\delta##, in an intuitive way for the specific case of the derivative.

To check my understanding: the concept of the derivative arises naturally when one poses the question, "what is the slope of a curve at a given point?". Obviously it doesn't make sense to calculate the slope at a single point as one needs two points along the curve to do so, and in particular, the ratio would be indeterminate (as we would have ##\frac{0}{0}##). Therefore, in order to define the rate of change in the value of a given function at a particular point we employ the properties of the limit. This is useful as ##\Delta x## can be allowed to approach zero, but not actually equal zero, and as such one can quantitatively define the derivative of a function at a given point ##x## as the limit, $$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ which equals the slope of the line tangent to the point ##x##. We therefore say that the rate of change in the value of the function ##f## with respect to a change in ##x## at a given point is defined as the limit given above, and is equal to the derivative of ##f## at ##x##, ##f'(x)##, if the limit exists.

Would this be correct at all?

 

[PeroK]

The beauty of mathematics is in its brevity, precision and unambiguity. You can describe the derivative in as many words as you like, but you can't say more than is expressed in those 10 or so symbols.

 

[Frank Castle]

The beauty of mathematics is in its brevity, precision and unambiguity. You can describe the derivative in as many words as you like, but you can't say more than is expressed in those 10 or so symbols.

Yes, that is true. That being said, I really want to check that I've understood the concept intuitively and I'm trying to describe as I currently understand it "in my head". Am I understanding it correctly at all?

 

[Mark44]

Sorry, very bad (and incorrect) phrasing on my part. I was trying to reword the definition of the limit, specifically the part ##0<\lvert\Delta x\rvert <\delta##, in an intuitive way for the specific case of the derivative.

Why bring in ##\delta##? That needlessly complicates things.

To check my understanding: the concept of the derivative arises naturally when one poses the question, "what is the slope of a curve at a given point?". Obviously it doesn't make sense to calculate the slope at a single point as one needs two points along the curve to do so, and in particular, the ratio would be indeterminate (as we would have ##\frac{0}{0}##). Therefore, in order to define the rate of change in the value of a given function at a particular point we employ the properties of the limit. This is useful as ##\Delta x## can be allowed to approach zero, but not actually equal zero, and as such one can quantitatively define the derivative of a function at a given point ##x## as the limit, $$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ which equals the slope of the line tangent to the point ##x##. We therefore say that the rate of change in the value of the function ##f## with respect to a change in ##x## at a given point is defined as the limit given above, and is equal to the derivative of ##f## at ##x##, ##f'(x)##, if the limit exists.

Would this be correct at all?

Looks fine to me, apart from what I said about the inequality with ##\delta##.

Yes, that is true. That being said, I really want to check that I've understood the concept intuitively and I'm trying to describe as I currently understand it "in my head". Am I understanding it correctly at all?

I don't see a problem.

 

[Frank Castle]

Looks fine to me, apart from what I said about the inequality with δδ\delta.

OK, thanks for all your help. I feel more confident in my understanding now!

One last thing. Is the definition of a derivative of a function at a point analogous to the definition of continuity? By this I mean, for continuity the statement is that a function ##f(x)## is said to be continuous at a point ##x=a## if the the limit of ##f(x)## as ##x## approaches ##a## exists and is equal to the value of ##f## at ##x=a##, ##f(a)##, i.e. $$\lim_{x\rightarrow a}f(x)=f(a)$$ For the derivative can we say that a function ##f(x)## is differentiable (has a derivative) at ##x=a## if the limit of ##\frac{f(a+\Delta x)-f(a)}{\Delta x}## as ##\Delta x## approaches 0 exists and is equal to the slope of the line tangent to the point ##x=a##, that is, the derivative of ##f## at ##x=a##, $$\lim_{\Delta x\rightarrow 0} \frac{f(a+\Delta x)-f(a)}{\Delta x}= f'(a)$$ I ask as it is not guaranteed that the value of $$\lim_{\Delta x\rightarrow 0} \frac{f(a+\Delta x)-f(a)}{\Delta x}$$ will be equal to the slope of the line tangent to the point ##x=a##, ##f'(a)##. Only when the function is differentiable at that point are they equal, right?!

 

[Mark44]

OK, thanks for all your help. I feel more confident in my understanding now!

One last thing. Is the definition of a derivative of a function at a point analogous to the definition of continuity? By this I mean, for continuity the statement is that a function ##f(x)## is said to be continuous at a point ##x=a## if the the limit of ##f(x)## as ##x## approaches ##a## exists and is equal to the value of ##f## at ##x=a##, ##f(a)##, i.e. $$\lim_{x\rightarrow a}f(x)=f(a)$$ For the derivative can we say that a function ##f(x)## is differentiable (has a derivative) at ##x=a## if the limit of ##\frac{f(a+\Delta x)-f(a)}{\Delta x}## as ##\Delta x## approaches 0 exists and is equal to the slope of the line tangent to the point ##x=a##, that is, the derivative of ##f## at ##x=a##, $$\lim_{\Delta x\rightarrow 0} \frac{f(a+\Delta x)-f(a)}{\Delta x}= f'(a)$$ I ask as it is not guaranteed that the value of $$\lim_{\Delta x\rightarrow 0} \frac{f(a+\Delta x)-f(a)}{\Delta x}$$ will be equal to the slope of the line tangent to the point ##x=a##, ##f'(a)##. Only when the function is differentiable at that point are they equal, right?!

If the latter limit exists, then the function f is differentiable at x = a, and the limiting value is f'(a). Your last sentence seems to imply that if a function isn't differentiable at x = a, then the limit could possibly exist but the slope of the tangent line at x = a (i.e., f'(a)) would have different values. That's not true. The derivative exists if and only if the limit exists.

For your first question here, about whether the definition of the derivative of f at a is analogous to the definition of continuity of f at a, I wouldn't say that they are analogous. If f is differentiable at a, it is automatically continuous at a, but not the other way around. A common example of a function that is continuous everywhere, but fails to have a derivative at one point is f(x) = |x|. Its derivative doesn't exist at x = 0 even though it is continuous there.

 

[Frank Castle]

f the latter limit exists, then the function f is differentiable at x = a, and the limiting value is f'(a). Your last sentence seems to imply that if a function isn't differentiable at x = a, then the limit could possibly exist but the slope of the tangent line at x = a (i.e., f'(a)) would have different values. That's not true. The derivative exists if and only if the limit exists.

Sorry, yes I've since thought about that remark and realized that it was incorrect (I was trying too hard to relate it to the definition of continuity - I read in a textbook somewhere that it is not necessarily true that the value of ##\lim_{x\rightarrow a}f(x)## is equal to the value ##f(a)##, only for continuous functions is this the case). So, in the case of the derivative, as you said, a function is differentiable at a point ##x=a## if the limit ##\lim_{\Delta x\rightarrow 0}\frac{f(a+\Delta x)-f(a)}{\Delta x}## exists. If this limit exists then its limiting value is exactly equal to the slope of the line tangent to the point ##x=a##, equivalently the rate of change in the value of the function ##f## with respect to a change in ##x##, ##f'(a)##.

Is the limit definition basically a method of deduction? That is, if we approach ##x=a## first from the left-hand side and then from the right-hand side and end up with the same limiting value for the function, or difference quotient (when considering the derivative), then we can deduce that this limiting value of the function as ##x## approaches ##x=a## is equal to the value of the (derivative of the) function at that point ##x=a##?

 

[Mark44]

Sorry, yes I've since thought about that remark and realized that it was incorrect (I was trying too hard to relate it to the definition of continuity - I read in a textbook somewhere that it is not necessarily true that the value of ##\lim_{x\rightarrow a}f(x)## is equal to the value ##f(a)##, only for continuous functions is this the case). So, in the case of the derivative, as you said, a function is differentiable at a point ##x=a## if the limit ##\lim_{\Delta x\rightarrow 0}\frac{f(a+\Delta x)-f(a)}{\Delta x}## exists. If this limit exists then its limiting value is exactly equal to the slope of the line tangent to the point ##x=a##

I think you still might be missing something. The derivative of a function f at a number a in the domain of f (i.e., f'(a)) is defined as the limit of the difference quotient. I infer from what you wrote just above, that the limit might possibly not exist, but that somehow the slope of the tangent line on the graph of f is meaningful. That is not the case. If f isn't differentiable at a, the graph doesn't have a tangent line at (a, f(a)).

, equivalently the rate of change in the value of the function ##f## with respect to a change in ##x##, ##f'(a)##.

Is the limit definition basically a method of deduction? That is, if we approach ##x=a## first from the left-hand side and then from the right-hand side and end up with the same limiting value for the function, or difference quotient (when considering the derivative), then we can deduce that this limiting value of the function as ##x## approaches ##x=a## is equal to the value of the (derivative of the) function at that point ##x=a##?

This paragraph doesn't make any sense to me. The limit definition of the derivative is a definition -- I don't see this as being connected with deduction at all. Also, for a limit to exist, both of the one-sided limits have to exist and be equal. This, too, is by definition.

Definitions are "if and only if" statements. If f'(a) exists, then the limit of the difference quotient has to exist and be equal to f'(a). If the limit of the difference quotient exists, then f'(a) has to exist and be equal to the limit of the difference quotient.

 

[Frank Castle]

This paragraph doesn't make any sense to me. The limit definition of the derivative is a definition -- I don't see this as being connected with deduction at all. Also, for a limit to exist, both of the one-sided limits have to exist and be equal. This, too, is by definition.

Definitions are "if and only if" statements. If f'(a) exists, then the limit of the difference quotient has to exist and be equal to f'(a). If the limit of the difference quotient exists, then f'(a) has to exist and be equal to the limit of the difference quotient.

I think I was trying to "look too much" into what the original reasoning might have been for the formulating the limit definition of the derivative and looking for deeper meanings that aren't there! Apologies for my ramblings!

I think you still might be missing something. The derivative of a function f at a number a in the domain of f (i.e., f'(a)) is defined as the limit of the difference quotient. I infer from what you wrote just above, that the limit might possibly not exist, but that somehow the slope of the tangent line on the graph of f is meaningful. That is not the case. If f isn't differentiable at a, the graph doesn't have a tangent line at (a, f(a)).

Sorry, this was badly worded by me, I simply meant that if the limit exists then the derivative of the function exists and this is equal to the slope of line tangent to the point in question, but of course this is true by definition so I was just being tautological :-\ . I get that the slope of the curve at a particular point has no meaning if the derivative doesn't exist. To be honest I was really trying to understand the statement I put in brackets about continuity. It seems incorrect as, like the limit definition of the derivative, continuity of a function at a point is defined in terms of the limit - a function is continuous at a given point if the limit of the function exists as we approach said point and that its limiting value is equal to the value of the function at that point. How is it possible to have a function whose limit (as defined above) exists, but that its limiting value is not equal to the value of the function at that point? (Because, if the limit exists then by definition it will equal the value of the function at that point)

 

[Mark44]

How is it possible to have a function whose limit (as defined above) exists, but that its limiting value is not equal to the value of the function at that point? (Because, if the limit exists then by definition it will equal the value of the function at that point)

No, this last part isn't true. Consider this function definition:
##f(x) = \begin{cases} 1 & x < 0 \\ 0 & x = 0 \\ 1 & x > 0 \end{cases}##
##\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = 1## but f(0) = 0.
Note that f is not continuous at x = 0.

 

[Frank Castle]

No, this last part isn't true. Consider this function definition:
##f(x) = \begin{cases} 1 & x < 0 \\ 0 & x = 0 \\ 1 & x > 0 \end{cases}##
##\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = 1## but f(0) = 0.
Note that f is not continuous at x = 0.

Of course, that example was staring me in the face, should've seen that. Thanks for highlighting it though!

Am I correct in saying that the definition of the derivative differs from the continuity definition (as discussed above) since the definition itself defines what the limiting value of the difference quotient (that one takes the limit of) is. That is, if the limit of the difference quotient, as ##x## approaches a given value, exists then its limiting value is, by definition, equal to the slope of the line tangent line to that point, i.e. the derivative of ##f(x)## at ##x=a## is defined as the limiting value ##f'(a)## of the difference quotient, nothing more nothing less. There is no case in which the limit of the difference quotient exists but is not equal to the slope, since this has no meaning (if the limit doesn't exist at a given point then there is no line that is tangent to the curve at that point and so the rate of change in the function cannot be defined at that point. In the continuous function case above, it does make sense to have the limiting value differ from the actual value of the function at a given point, since discontinuous functions exist, and the existence of the value of the function itself, at a given point, does not depend on the limit. (Sorry the last bit in brackets is a bit convoluted and maybe misworded.)

 

[Mark44]

Of course, that example was staring me in the face, should've seen that. Thanks for highlighting it though!

Am I correct in saying that the definition of the derivative differs from the continuity definition (as discussed above) since the definition itself defines what the limiting value of the difference quotient (that one takes the limit of) is. That is, if the limit of the difference quotient, as ##x## approaches a given value, exists then its limiting value is, by definition, equal to the slope of the line tangent line to that point, i.e. the derivative of ##f(x)## at ##x=a## is defined as the limiting value ##f'(a)## of the difference quotient, nothing more nothing less.

No.
For a function that is differentiable at a:
##\lim_{h \to 0}\frac{f(a + h) - f(a)}{h} = f'(a)##
For a function that is continuous at a:
##\lim_{h \to 0} f(a + h) = f(a)##

The important distinction between continuity and differentiability is that the latter is a stronger condition: every function that is differentiable at a is automatically continuous at a. A function that is merely continuous at a does not have to be differentiable there; for example, f(x) = |x|.

There is no case in which the limit of the difference quotient exists but is not equal to the slope, since this has no meaning (if the limit doesn't exist at a given point then there is no line that is tangent to the curve at that point and so the rate of change in the function cannot be defined at that point. In the continuous function case above, it does make sense to have the limiting value differ from the actual value of the function at a given point, since discontinuous functions exist, and the existence of the value of the function itself, at a given point, does not depend on the limit. (Sorry the last bit in brackets is a bit convoluted and maybe misworded.)

If you are referring to the piecewise-defined example I gave earlier, what you said here is incorrect. For a function to be continuous at a, it must be true that ##\lim_{x \to a}f(x)## is equal to f(a).

 

[Frank Castle]

For a function that is differentiable at a:
limh→0f(a+h)−f(a)h=f′(a)limh→0f(a+h)−f(a)h=f′(a)\lim_{h \to 0}\frac{f(a + h) - f(a)}{h} = f'(a)
For a function that is continuous at a:
limh→0f(a+h)=f(a)limh→0f(a+h)=f(a)\lim_{h \to 0} f(a + h) = f(a)

The important distinction between continuity and differentiability is that the latter is a stronger condition: every function that is differentiable at a is automatically continuous at a. A function that is merely continuous at a does not have to be differentiable there; for example, f(x) = |x|.

Right. Yes, I understand that. What I meant was that ##f'(a)## is defined as the limiting value of ##\frac{f(a+h)-f(a)}{h}## in the limit as ##x## tends to ##a## (##h## tends to zero). It is the limit that defines what the value of the derivative is at a given point, whereas for the continuity case, if ##f(x)## is continuous at ##x=a## then it's limiting value as ##x## tends to ##a## (##h## tends to zero) is equal to ##f(a)##, i.e. ##\lim_{h\rightarrow 0}f(a+h)=f(a)##. However, if ##f## is discontinuous at a point, then the limit ##\lim_{x\rightarrow a}f(x)## can exist, but it's limiting value won't be equal to ##f(a)##, as shown in your example.
What I was trying to highlight was that if the limit ##\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}## exists, then by definition it's limiting value is ##f'(a)##, whereas the limit ##\lim_{x\rightarrow a}f(x)## can exist and have some limiting value, but this isn't necessarily equal to ##f(a)## (of course, if ##f## is continuous, then by definition it is equal), in other words, the value ##f(a)## is not defined as the limiting value of the limit ##\lim_{x\rightarrow a}f(x)##, but when it is equal to the limiting value of this limit, then the function is continuous at that point.

 

[Mark44]

However, if ##f## is discontinuous at a point, then the limit ##\lim_{x\rightarrow a}f(x)## can exist, but it's limiting value won't be equal to ##f(a)##, as shown in your example.

Another possibility is that this limit doesn't exist at all, such as for f(x) = 1/x, with a = 0.

 

[Frank Castle]

Another possibility is that this limit doesn't exist at all, such as for f(x) = 1/x, with a = 0.

True, good point.

Would the rest of what I put in my last post be correct? I think I'm understanding the derivative definition correctly now, would you say that's fair?

 

[dustball]

Undergraduate level (I'm ashamed to say). I understand how to use the derivative and its associated rules, etc. and I thought I understood intuitively what a derivative is, but now I'm not so sure, the description I read on Wikipedia has really thrown a spanner in the works for me (I'm not happy with simply accepting definitions when I learn maths, I want to have an intuitive idea of what the particular operation is describing). What the sticking point for me is, how do I make sense of the statement:
What exactly does this statement mean? Is it simply that although the function has a fixed value at that point the value of the function will in general be changing from point to point and the derivative evaluated at a particular point describes the rate at which its value is changing at that point in the sense that if a move away from that point (say ##x_{0}##) by a small amount ##\Delta x##, the value of the function ##f## will approximately change by an amount ##f'(x_{0})\Delta x##?

Correct.

 

[rrogers]

To give an alternate non-calculus viewpoint we can use a very old fashioned statement. The derivative is the slope of the line/plane that intercepts a curve/surface at exactly one-point. If the line isn't unique then the "derivative" isn't unique. It is presumed that the line is as short as necessary to avoid bumps. This leads into the whole idea that tangents/derivatives can be said to live in a different tangent space associated with a curve and not related to the space the surface/curve is defined in.
This is from some very old books that tried to avoid mensuration and wanted "intrinsic" definitions (I presume).
In addition: a derivation is defined by ## D(f(x)*g(x))=f(x)*D(g(x))+D(f(x))*g(x) ##
I add this to show that the derivative operator is a member of a broader class of operators having specific properties.