The second derivative

Differentiation: Applications of derivatives

The second derivative

The derivative $f'$ of a function $f$ can be derived again. We call this the second derivative of $f$ .

For a function $\blue{f(x)}$ , we denote the second derivative as:

$\green{f''(x)}=\frac{\dd}{\dd x}\orange{f'(x)}=\frac{\dd}{\dd x}\left(\frac{\dd}{\dd x}\blue{f(x)}\right)$

Example

$\begin{array}{rcl}\blue{f(x)}&\blue{=}&\blue{3x^2}\\ \orange{f'(x)}&\orange{=}&\orange{6x}\\\green{f''(x)}&\green{=}&\green{6}\end{array}$

The second derivative is useful when one wants to find the extreme values of a function $f(x)$ . We saw earlier that the condition $f'(c)=0$ did not immediately imply that $c$ corresponds to an extreme value. The following theorem will help us determining whether such a value corresponds to an extreme value or not.

If for a function $\blue{f(x)}$ and a point $x=\purple{c}$ we have

$\orange{f'(}\purple{c}\orange{)}=0$
$\green{f''(}\purple{c}\green{)}\neq 0$ ,

then, $\blue{f(x)}$ has an extreme value in $\purple{c}$ .

If $\green{f''(}\purple{c}\green{)}>0$ , then $\purple{c}$ corresponds to a local minimum. If $\green{f''(}\purple{c}\green{)}<0$ , then $\purple{c}$ corresponds to a local maximum.

Example

$\begin{array}{rcl}\blue{f(x)}&=&\blue{2x^2+x}\\ \orange{f'(x)}&=&\orange{4x+1}\\ \green{f''(x)}&=&\green{4}\\ \orange{f'(}\purple{-\frac{1}{4}}\orange{)}&=&0\\ \green{f''(}\purple{-\frac{1}{4}}\green{)}&=&4\neq 0\end{array}$

In the example, $\blue{f(}\purple{-\frac{1}{4}}\blue{)}=\frac{3}{8}$ is a local minimum of $\blue{f(x)=2x^2+x}$ .

This theorem does not hold if $\purple{c}$ is a boundary value of the domain $\blue{f(x)}$ is defined on.

The step-by-step method of finding the extreme values of a function $f(x)$ gets much easier using this theorem. We can replace step $3$ by just checking if the found zeroes are smaller, larger or equal to $0$ .

Calculate $f''(x)$ for $f(x)=6\cdot x^4-2\cdot x^2+7$ .

Simplify your answer as much as possible.

$f''(x)=$ $72\cdot x^2-4$

We first calculate the first derivative using the power rule.

$f'(x)=24\cdot x^3-4\cdot x$
Then we calculate the second derivative in the same way.

$f''(x)=72\cdot x^2-4$

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