Thread: Logarithmic Differentiation

4. e) Use Logarithmic differention to find the derivative of the function.

f(x) = (cos x)^x

I can turn it into x ln cos x but I do not know how to finish this. Please help.

f(x) = (cos x)^x

lnf(x)=ln{(cos x)}^x

lnf(x)=x*ln(cos x)

Now we derivative according to x.

f'(x)/f(x) = ln(cos x) + x*{(-sin x)/(cos x))

f'(x) = f(x)*{ln(cos x) + x*{(-sin x)/(cos x))}

f'(x)= {(cos x)^x}*{ln(cos x) + x*{(-sin x)/(cos x))}

Take the ln of both sides.



Differentiate both sides with respect to x.



Use the chain rule on the left hand side and the product rule on the RHS.



Now use the chain rule on the RHS.



Finally multiply both sides by 'y'.

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