Let $\displaystyle Y$ be a random variable and denote its pdf $\displaystyle f_Y$

In general, let $\displaystyle g $ be an increasing function on the range of $\displaystyle Y$. Define $\displaystyle W=g^{-1}(Y)$. Show that $\displaystyle f_W(w) = f_Y(g(w))\cdot g^{\prime}(w)$

My question is how to incorporate the fact that g is an increasing function on the range of Y into my proof. I believe it belongs when I mess with the inverse, but I'm not sure how to say how I use that fact correctly.

Work:

$\displaystyle F_W(w) = P(W<w) = P(g^{-1}(Y)<w) = P(Y<g(w)) = F_Y(g(w))$

$\displaystyle f_W(w)=\frac{d}{dw}F_W(w)=\frac{d}{dw}F_Y(g(w))=f_ Y(g(w))\cdot g^{\prime}(w)$