Function Type:

What is the approximate value of the derivative at $$x=$$ ?

## About Graphing Derivatives

The derivative $$f'(x)$$ of a function $$f(x)$$ is defined as the limit $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ At a particular value $$a$$, we can also define $$f'(a)$$ as $f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$ if this limit exists. This limit will be equal to the slope of the tangent line to $$f(x)$$ at $$x=a$$. By approximating the value of $$f'(a)$$ for several values of $$a$$, we can sketch a graph of $$f'(x)$$, without having to compute it.

## Using the Applet

Choose a type of function from the pull-down menu. The graph of the function will be drawn, and several values along the $$x$$-axis will be chosen. For each $$x$$, approximate the value of $$f'(x)$$. Remember, this is the slope of the line tangent to the graph of the function at $$x$$. You can choose to display each tangent line to assist. If you choose the correct answer, a point will be plotted at $$(x,f'(x))$$. If an incorrect answer is chosen, you'll have to answer correctly before moving on to the next value of $$x$$. Once all values have been correctly found, the derivative will be drawn.