Mathematics

Problem

Let

g(x)

be a differentiable function such that

g(x)\ne 0

for all

x

in its domain. Answer the following: a. (i) Starting with

y=\ln|g(x)|

, derive an expression for

\dfrac{dy}{dx}

in terms of

g(x)

and

g'(x)

. b. (ii) Hence, obtain the general form of

\displaystyle\int \dfrac{g'(x)}{g(x)}\,dx

. c. (iii) Taking

g(x)=\sin x

, evaluate

\displaystyle\int \dfrac{\cos x}{\sin x}\,dx

and verify your answer by differentiation.