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: $$ f(x) = x g(x)^{-\frac{1}{2}} $$ 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)