Sketching the derivative

Recall that $f'$ is the function or formula with which we can calculate the slope of the tangent to the graph of a given function $f$:

Both $f$ and $f'$ are functions, and we can draw their graphs (given that we have their algebraic expressions). Here is an example:

Example 1

Consider the function $f(x)=x^2$. The derivative is $f'(x)=2x$.

$$\begin{array}{cllc} x & & & x &\\ \large\downarrow & & & \large\downarrow & \\ \boxed{\large f} & \text{rule: } x^2 & \Longrightarrow & \boxed{\large f'} & \text{rule: } 2x\\ \large\downarrow & & & \large\downarrow & \\ y & & & y & \\ \end{array}$$

And we can draw their graphs as usual. Note that the height of the graph $f'$ at $x$ is just the slope of the tangent to $f$ at $x$.

So, given a graph of a function $f$, it is no problem to sketch the graph of $f'$:

• select a point $x$ on the $x$-axis.
• draw the tangent to the graph of $f$ at this point.
• estimate the slope of this tangent (you can use the slope triangle), this is an estimate of the height of the graph of $f'$ at $x$.

If you do this for several positions $x$, we can sketch the graph of $f'$. An example is shown below.

Example 2

A graph of a function $f$ is given (upper graph in the figure below). Use this graph to sketch the graph $f'$. Note:

• Often it is useful to start with the horizontal tangents, as these are the $x$-intercepts of $f'$, and easy to find.
• Then choose a couple of points between these horizontal tangents and estimate their slope (for example with the slope triangle). It does not have to be super exact.