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

##### 1 Answer

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 '( * ).

#d/dx(ln(cos(x)))=1/(cos(x)) * d/dx(cos(x))#

#= (1)/(cos(x))*(-sin(x))#

#=(-sin(x))/cos(x)=-tan(x).#

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

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