If f is an always positive, differentiable function and y = ln(f(sin x)), what is y’ when x=4 ?

cos(4) f'(sin 4)/f(sin 4) is y' at x=4

Step-by-step explanation:

We can't actually find the numerical value for this without more information.

We are given:

y=ln(f(sin x))

This is equivalent to

e^y=f(sin x)

Differentiate both sides:

y' e^y=cos(x) f'(sin x)

Divide both sides by e^y:

y' =cos(x) f'(sin x)/e^y

Rewrite in terms of x using that e^y=f(sin x):

y' =cos(x) f'(sin x)/f(sin x)

Now replace x with 4:

cos(4) f'(sin 4)/f(sin 4) is y' at x=4