you're having trouble with the chain rule
as I wrote above you CANNOT change the inner function g(x)
In the outer function is ln and the inner is cosine
The derivative of the log is 1 over the ENTIRE inner function giving us
Next multiply that by the derivative of the inner function
the derivative of the cosine x is .
Next multiply these two.
But I see how you did it. 1 over the inner function left untouch. I'm struggling to understand why ends up within the differentiation of unless you replace the in the function with the inner function.
So basically, any would produce a solution with where the inner function will always end up at the bottom replacing e.g.
is that correct?
BTW, thank you for your help
you did change the inner function, which is cosine x not just x.
The chain rule is just the product of two functions f' of g(x) which is one over g(x)=cos x here, then times the derivative of cos x.
Try setting f(x)=ln x and g(x) =cos x. Obtain the derivatives and plug into f'(g(x)) and g'(x).
and yes, the derivative of ln sin x is cot x.
More importantly you now know the integral of tan x and cot x.