# Math 22 Chain Rule

## The Chain Rule

 If ${\displaystyle y=f(x)}$ is a differentiable function of ${\displaystyle u}$ and ${\displaystyle u=g(x)}$ is a
differentiable function of ${\displaystyle x}$, then ${\displaystyle y=f(g(x))}$ is a differentiable function
of ${\displaystyle x}$ and

${\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}}$

In another word, ${\displaystyle {\frac {d}{dx}}[f(g(x))]=f'(g(x))\cdot g'(x)}$


Example: Find derivative of

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

Solution:
${\displaystyle f(x)=f(x)={\sqrt {x^{2}+3x-4}}=(x^{2}+3x-4)^{\frac {1}{2}}}$
${\displaystyle f'(x)={\frac {1}{2}}\cdot (x^{2}+3x-4)^{({\frac {1}{2}}-1)}{\frac {d}{dx}}[x^{2}+3x-4]}$
${\displaystyle =(x^{2}+3x-4)^{\frac {-1}{2}}(2x+3)}$

2) ${\displaystyle f(x)=(x^{2}+1)^{100}}$

Solution:
${\displaystyle f'(x)=100(x^{2}+1)^{99}{\frac {d}{dx}}[x^{2}+1]}$
${\displaystyle =100(x^{2}+1)^{99}(2x)}$

## The General Power Rule

 If ${\displaystyle y=[u(x)]^{n}}$, where ${\displaystyle u}$ is a differentiable function of ${\displaystyle x}$
and ${\displaystyle n}$ is a real number, then

${\displaystyle {\frac {d}{dx}}[u^{n}]=n\cdot u^{n-1}\cdot u'}$