Suppose $X$ is a r.v with pdf $f_X(x)$. Let $Y = g(X)$. To find the pdf of $Y$ - $f_Y(y)$. I use one of two ways and I assume g to be a monotonically increasing function.
Method I
first using the standard change of variable in integration
Since, $dy = g'(x)dx$
$f_X(x) dx = f_X(g^{-1}(y)) \frac{dy}{g'(g^{-1}(y))}$. From this I infer that
(1) $f_Y(y) = \frac{f_X(g^{-1}(y))}{g'(g^{-1}(y))}$
Method II
Consider the distribution function of $Y$, $F_{Y}(x)$
$F_{Y}(y) = \mathbb{P}(Y < y) = \mathbb{P}(X < g^{-1}(y)) = F_{X}(g^{-1}(y))$
Now to find pdf of $Y$. I differentiate the distribution function with respect to y and get,
(2) $\frac{d}{dy} F_{Y}(y) = f_{Y}(y) = f_{X}(g^{-1}(y)).\frac{d}{dy}g^{-1}(y)$
Q1) My question is that even for monotonically increasing function where method II is applicable (1) and (2) seem to differ. Why?
Q2) Does Method I always yield the right answer for any type of function?