User:Nsh/Chain-rule question

The original problem set is: $f(x)=\frac{x}{\sqrt{3-x}}$

There are two techniques of differentiation required to solve the problem: Chain_rule and Product_rule
The chain rule because we have a funtion of a function: $f(x)=\frac{x}{\sqrt{g(x)}}$ where $g(x)=3-x$

And the product rule because we are effectivly differentiating a product: $f(x)=xg(x{)}^{-\frac{1}{2}}$

By the chain rule ( $\frac{d}{dx}f(g(x))=\frac{d}{dx}g\frac{d}{dx}f$ ) we can use the substitution $u=3-x$
$du/dx=-1$
f`(x) = $\frac{du}{dx}x(-\frac{1}{2})x^{-\frac{3}{2}}+\frac{x}{\sqrt{3-x}}=\frac{1}{2}{x}^{-\frac{1}{2}}+\frac{x}{\sqrt{3-x}}$ (Two terms because of Product_rule)