Mercyhurst UniversityMath DeptDr Williams Home



x = 1

Show Tangent Line at x
First Derivative
Second Derivative

About Derivatives

The first 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 \(x=a\).

The second derivative \(f''(x)\) is found by taking the derivative of the first derivative. That is, the value of \(f''(x)\) is the slope of the tangent line to the graph of \(f'(x)\) at \(x\).

Using the Applet

Choose a function to work with, and use the slider to adjust the current value of \(x\). The tangent line to \(f(x)\) at \(x\) will be displayed, along with the graphs of \(f'(x)\) and \(f''(x)\) if these options are enabled.

About this Applet

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.