Question

How do you differentiate `(ln(x+1))^(cosx)`?

Answer 1

`f'(x)=(ln(x+1))^cos(x)*(-sin(x)*ln(ln(x+1))+cos(x)/((x+1)*ln(x+1))`

Explanation:

To make thngs better we take the logaithm on both sides:
`ln(f(x))=cos(x)ln(ln(x+1))`
Now using the chain rule to differentiate both sides with respect to `x`:

`1/f(x)*f'(x)=-sin(x)ln(ln(x+1))+cos(x)*1/ln(x+1)*1/(x+1)`

multiplying by
`f(x)`

`f'(x)=(ln(x+1))^cos(x)*(-sin(x)*ln(ln(x+1))+cos(x)/((x+1)*ln(x+1)))`