# What is the first derivative and second derivative of x^4 - 1?

May 31, 2016

${f}^{'} \left(x\right) = 4 {x}^{3}$
${f}^{'} ' \left(x\right) = 12 {x}^{2}$

#### Explanation:

to find the first derivative we must simply use three rules:

1. Power rule
$\frac{d}{\mathrm{dx}} {x}^{n} = n {x}^{n - 1}$

2. Constant rule
$\frac{d}{\mathrm{dx}} \left(c\right) = 0$ (where c is an integer and not a variable)

3. Sum and difference rule
$\frac{d}{\mathrm{dx}} \left[f \left(x\right) \pm g \left(x\right)\right] = \left[{f}^{'} \left(x\right) \pm {g}^{'} \left(x\right)\right]$

the first derivative results in:
$4 {x}^{3} - 0$
which simplifies to
$4 {x}^{3}$

to find the second derivative, we must derive the first derivative by again applying the power rule which results in:
$12 {x}^{3}$

you can keep going if you like:
third derivative = $36 {x}^{2}$
fourth derivative = $72 x$
fifth derivative = $72$
sixth derivative = $0$