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Drawing the derivative of a graph

Petrus

Well-known member
Feb 21, 2013
739
Hello,

I have been having problems drawing the derivative of a function. What I mean is that just given the graph of some function $f$ and not its definition, you are supposed to draw $f'$. I understand that when the tangent line of the graph is horizontal, this will correspond to $f'(x)=0$, but my question is if I draw all the extrema points how can I know what the rest of graph will look like?

Well, to make this more clear, let's say I give you the graph of $f(x)=x^3-x$ (I give you the graph, but you don't know its actual definition). I can see that the graph changes from going up to down around $x=-0.8$ (I don't know if it's called a max point in English) and the function then changes from going down to up around $x=0.8$ (I don't know if it's called a min point in English) that means I know the two roots of the derivative, but how can i draw the rest of the derivative's graph?
 
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chisigma

Well-known member
Feb 13, 2012
1,704
Re: Draw f' of a graph

Hello,
i have been having problem to draw the derivate of a function, what i mean that u just get the graph and not the function and u suposed to draw f'. I understand that when the horizontal line of the graph of the function Will give u the point when y=0 at the x point but My question is if i draw all the y=0 how can i know how the rest of graph Will look like?
Well to make this more Clear lets say i give u the graph f(x)=x^3-x (i give u that graph and u dont know its the function f(x)-x^3-x. I can se that the graph change from going up to down in x=-0.8( idk if it Calle Max point in english) and the function go down to up in x=0.8 ( idk if it calls min point in english)that means it Will be My y=0 how can i draw the rest?
If only the graph of the function is allowable [that happens for example when the function represents experimental data...] the best for You probably is to approximate the function with a polynomial of degree n [i.e. last square polynomial approximation...] that can easily be derived in a successive step...


Kind regards


$\chi$ $\sigma$
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Draw f' of a graph

Petrus,

I took the liberty of editing your post to make it a bit easier to read. (Smile)

You are starting at a good place with the type of problem you describe, that is, to first identify that the extrema of the function will correspond to the roots of the derivative, since these point are where the instantaneous rate of change of the function is zero.

Next, look at the intervals where the function is increasing. On these intervals, the graph of the derivative will be positive. Likewise, on the intervals where the function is decreasing, the derivative will be negative. The more rapidly the function is changing, the further the graph of the derivative will be from the $x$-axis.