# What is the derivative of f(x)=ln(cos(x)) ?

Aug 10, 2014

In f(x) = ln(cos(x)), we have a function of a function (it's not multiplication, just sayin'), so we need to use the chain rule for derivatives:

d/dx(f(g(x)) = f'(g(x))*g'(x)

For this problem, with f(x) = ln(x) and g(x) = cos(x), we have f '(x) = 1/x and g'(x) = - sin(x), then we plug g(x) into the formula for f '( * ).

$\frac{d}{\mathrm{dx}} \left(\ln \left(\cos \left(x\right)\right)\right) = \frac{1}{\cos \left(x\right)} \cdot \frac{d}{\mathrm{dx}} \left(\cos \left(x\right)\right)$
$= \frac{1}{\cos \left(x\right)} \cdot \left(- \sin \left(x\right)\right)$
$= \frac{- \sin \left(x\right)}{\cos} \left(x\right) = - \tan \left(x\right) .$

This is worth remembering for later when you learn about integrals!