# How do you differentiate f(x)=csc(ln(1/x^4))  using the chain rule?

May 21, 2017

$\frac{4 \csc \left(4 \ln x\right) \cot \left(4 \ln x\right)}{x}$

#### Explanation:

This can be rewritten as:
$- \csc \left(4 \ln x\right)$

The derivative of:
$\csc \left(u\right) = - \csc \left(u\right) \cot \left(u\right)$
$\ln u = \frac{1}{u}$

Solving by the chain rule:
$\frac{d}{\mathrm{dx}} \left(- \csc \left(4 \ln x\right)\right) = \csc \left(4 \ln x\right) \cot \left(4 \ln x\right) \left(\frac{d}{\mathrm{dx}} \left[4 \ln x\right]\right)$
$= \csc \left(4 \ln x\right) \cot \left(4 \ln x\right) \left(\frac{4}{x}\right)$
$= \frac{4 \csc \left(4 \ln x\right) \cot \left(4 \ln x\right)}{x}$