Finding the Slope of a Function: A Scientific Approach

[TommG]

I have to find the slope of the function

g(x) = x/(x-2), (3,3)

my attempt

[(3+h)/((3+h)-2)] - [(3)/(3-2)] [itex]\div[/itex] h

got rid of (3+h) and 3
[(1/-2) -(1/-2)] [itex]\div[/itex] h
0/h

answer in book is -2

 

[adjacent]

I have to find the slope of the function

g(x) = x/x-2, (3,3)

my attempt

[(3+h)/((3+h)-2)] - [(3)/(3-2)] [itex]\div[/itex] h

got rid of (3+h) and 3
[(1/-2) -(1/-2)] [itex]\div[/itex] h
0/h

answer in book is -2

##g(x)=\frac{x}{x-2}## right?
Can't you use the quotient rule?

 

[TommG]

##g(x)=\frac{x}{x-2}## right?
Can't you use the quotient rule?

yes g(x)= x/(x-2)

I don't think I can use the quotient rule. Don't you need a limit? I wasn't given a limit only a function.

 

[adjacent]

yes g(x)= x/(x-2)

I don't think I can use the quotient rule. Don't you need a limit? I wasn't given a limit only a function.

No.
You can solve it in two ways.

1. The definition of the derivative of a function
2. Quotient rule

The definition of the derivative of a function ##f## with respect to x: ##\frac{\text{d}f}{\text{d}x}=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}##

The quotient rule states that ##(\frac{u}{v})'=\frac{u'v-uv'}{v^2}## Where u is the numerator and v is the denominator.

The quotient rule is much easier.

 

[verty]

I have to find the slope of the function

g(x) = x/(x-2), (3,3)

my attempt

[(3+h)/((3+h)-2)] - [(3)/(3-2)] [itex]\div[/itex] h

got rid of (3+h) and 3
[(1/-2) -(1/-2)] [itex]\div[/itex] h
0/h

answer in book is -2

##{x \over x-2} \not = {1 \over -2}##.

You may want to review fractions and what manipulations are allowed.

 

[TommG]

No.
You can solve it in two ways.

1. The definition of the derivative of a function
2. Quotient rule

The definition of the derivative of a function ##f## with respect to x: ##\frac{\text{d}f}{\text{d}x}=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}##

The quotient rule states that ##(\frac{u}{v})'=\frac{u'v-uv'}{v^2}## Where u is the numerator and v is the denominator.

The quotient rule is much easier.

Ok then I have to use the first option. Not allowed to use the second option yet.

this is the definition I have to use

The definition of the derivative of a function ##f## with respect to x: ##\frac{\text{d}f}{\text{d}x}=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}##

 

[adjacent]

I have to find the slope of the function

g(x) = x/(x-2), (3,3)

my attempt

[(3+h)/((3+h)-2)] - [(3)/(3-2)] [itex]\div[/itex] h

got rid of (3+h) and 3
[(1/-2) -(1/-2)] [itex]\div[/itex] h
0/h

answer in book is -2

Your attempt:$$\frac{\frac{3+h}{(3+h)-2}-\frac{3}{3-2}}{h}$$ is correct.
You only have some algebra problems in simplifying that. Try again.

If you use that definition,you will get the derivative with respect to x. If you use 3 instead of x, you will get derivative of the function at x=3. That's what you did there in your attempt

 

[TommG]

thank all of you who helped I have figured it out. I do not need help anymore.

 

[adjacent]

thank all of you who helped I have figured it out. I do not need help anymore.

Happy to know that