Question

Use logarithmic differentiation to prove D9: d/dx (1/u^n) =( -n/u^n+1)(du/dx)

Answer 1

$f(x)=\frac{1}{u^n}$

$\ln{f(x)}=\ln{\frac{1}{u^n}}=-n\ln u$

$\frac{d}{dx}\ln{f(x)}=\frac{d}{dx}(-n\ln u)$

$\frac{1}{f(x)}\frac{df(x)}{dx}=-n\frac{1}{u}\frac{du}{dx}$

$\frac{df(x)}{dx}=-nf(x)\frac{1}{u}\frac{du}{dx}=-n\frac{1}{u^{n+1}}\frac{du}{dx}$

$\frac{d}{dx}\frac{1}{u^n}=-n\frac{1}{u^{n+1}}\frac{du}{dx}$