6B. Second Derivatives
Instructions:
- Use the input box to define the function f(x).
- Use the checkboxes for f(x), f'(x), and f''(x) to show/hide the graphs of the function, its first derivative, and its second derivative, respectively.
- Use the Tangent checkboxes to show/hide tangent lines on the graph of f and the graph of f', respectively.
The Second Derivative
Given a function y = f(x), the derivative f'(x) is a function whose values represent the rate of change of f. Because y = f'(x) is itself a function, it has its own rate of change, i.e., derivative. The derivative of the derivative of f is called the second derivative of f:
Because the derivative operator is applied to y = f(x) twice, the following notation is also frequently used to represent the second derivative:
Whereas the first derivative tells us how fast f is changing and can be thought of like speed/velocity of f, the second derivative tells us how fast f' is changing and can be thought of like acceleration of f.