# Second Derivative

How to find the second derivative of a function: definition, 1 example, and its solution.

## Definition

The second derivative is the derivative of a derivative.

So, to find the second derivative,

differentiate the function [twice].

It is written as f''(x), y'', d^{2}y/dx^{2}, (d^{2}/dx)f(x).

## Exampley = x^{5} - 7x^{2} - 8x + 1, f''(x) = ?

Solution

Solution (Detail)

First find the derivative f'(x).

f(x) = x^{5} - 7x^{2} - 8x + 1.

Then f'(x) = 5x^{4} - 7⋅2x^{1} - 8 + 0.

Derivative of a Polynomial

-7⋅2x^{1} = -14x

f'(x) = 5x^{4} - 14x - 8.

To find f''(x), differentiate again:

find the derivative of f'(x).

Then f''(x) = 5⋅4x^{3} - 14 - 0.

5⋅4x^{3} = 20x^{3}

So f''(x) = 20x^{3} - 14.