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) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)={\sqrt {x^{2}+3x-4}}}

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=f(x)=\sqrt{x^2+3x-4}=(x^2+3x-4)^{\frac{1}{2}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\frac{1}{2}\cdot (x^2+3x-4)^{\frac{1}{2} -1}\frac{d}{dx}[x^2+3x-4]}
<math=(x^2+3x-4)^{\frac{-1}{2}}(2x+3)

2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=(x^2+1)^100}

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=100(x^2+1)^99 \frac{d}{dx}[x^2+1]}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =100(x^{2}+1)^{9}9(2x)}

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