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.

About this Applet

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