Question

What is the derivative of `f(x) = ln(sinx))`?

Answer 1

`f'(x)=cotx`

Explanation:

We'll apply the Chain Rule, which, when applied to logarithms, tells us that if `u` is some function in terms of `x,` then

`d/dxlnu=1/u*(du)/dx`

Here, we see `u=sinx,` so

`f'(x)=1/sinx*d/dxsinx`

`f'(x)=cosx/sinx`

`f'(x)=cotx`

Answer 2

`f'(x)=cotx`

Explanation:

`"differentiate using the "color(blue)"chain rule"`

`"Given "f(x)=g(h(x))" then"`

`f'(x)=g'(h(x))xxh'(x)larrcolor(blue)"chain rule"`

`"here "f(x)=ln(sinx)`

`rArrf'(x)=1/sinx xxd/dx(sinx)=cosx/sinx=cotx`