# Math 22 Higher-Order Derivative

## Higher-Order Derivatives

``` The "standard" derivative ${\displaystyle f'(x)}$ is called the first derivative of ${\displaystyle f(x)}$. The derivative of ${\displaystyle f'(x)}$ is the second derivative of${\displaystyle f(x)}$, denoted by ${\displaystyle f''(x).}$
By continuing this process, we obtain higher-order derivative of ${\displaystyle f(x)}$.
```

Note: The 3rd derivative of ${\displaystyle f(x)}$ is ${\displaystyle f'''(x)}$. However, we simply denote the ${\displaystyle n^{th}}$ derivative as ${\displaystyle f^{(n)}(x)}$ for ${\displaystyle n\geq 4}$

Example: Find the first four derivative of

1) ${\displaystyle f(x)=x^{4}+5x^{3}-2x^{2}+6}$

Solution:
${\displaystyle f'(x)=4x^{3}+15x^{2}-4x}$
${\displaystyle f''(x)=12x^{2}+30x-4}$
${\displaystyle f'''(x)=24x+30}$
${\displaystyle f^{(4)}(x)=24}$

2) ${\displaystyle f(x)=(x^{3}+1)(x^{2}+3)}$

Solution:
It is better to rewrite ${\displaystyle f(x)=(x^{3}+1)(x^{2}+3)=x^{5}+3x^{3}+x^{2}+3}$
Then, ${\displaystyle f'(x)=5x^{4}+9x^{3}+2x}$
${\displaystyle f''(x)=20x^{3}+27x^{2}+2}$
${\displaystyle f'''(x)=60x^{2}+54x}$
${\displaystyle f^{(4)}(x)=120x+54}$

## Notes

If ${\displaystyle f(x)}$ is the position function, then ${\displaystyle f'(x)}$ is the velocity function and ${\displaystyle f''(x)}$ is the acceleration function.