Math 22 Chain Rule

The Chain Rule

 If $y=f(x)$ is a differentiable function of $u$ and $u=g(x)$ is a 
 differentiable function of $x$, then $y=f(g(x))$ is a differentiable function 
 of $x$ and
 
 ${\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}$
 
 In another word, ${\frac {d}{dx}}[f(g(x))]=f'(g(x))\cdot g'(x)$

Example: Find derivative of

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

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

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

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

The General Power Rule

 If $y=[u(x)]^{n}$, where $u$ is a differentiable function of $x$ 
 and $n$ is a real number, then
 
 ${\frac {d}{dx}}[u^{n}]=n\cdot u^{n-1}\cdot u'$

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