Density of a one-to-one function real life example If $Y$ is a function of the random variable $X$ with probability density function $f_x(x)$ where $y=g(x)$; then the density function of $Y$ is 
$$
f_Y(y)= f_X(g^{-1}(y)) \left|  \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d} y} \right|
$$
This is one of my favorite probability tricks - I think mapping one domain to another is the basis for so many algorithms, etc. However, I have never seen, or perhaps never recognized, a real life example of this theorem. 
If someone is aware of a real life application and can show a little bit of math with it - I would really appreciate it. 
 A: In a real data analysis problem I was involved in, the question of whether to use a lognormal model (take logs and fit a linear regression) or a gamma GLM with log-link was being discussed (to fit some insurance data) - and how much difference it might make to the results for the problem at hand.
Naturally enough, the question of what the effect of taking logs (and ultimately fitting a model) would be if the distribution were really Gamma arose.
Consequently, the question of "what's the density of the log of a Gamma random variable" was one of the first things to consider, and the formula you quote can be used to answer that question (though I think in that case I just did a direct change of variable by hand rather than plugging in the formula, I could have used the formula to the same end). I had answered this question some time before, though, so I already knew more-or-less what the answer looked like.
As requested in comments -- here's hoping I don't make any errors, I'm not going to have time to check this at the moment, nor even work carefully (so caveat emptor, this is probably worth about what you paid for it):
A variable, $X\sim \text{Gamma}(\alpha,\beta)$ (for a scale parameterization rather than a rate parameterization) has density 
$f_X(x) =\begin{cases} \frac{1}{\Gamma(\alpha) \beta^\alpha} x^{\alpha-1} e^{-x/\beta};\, & x>0, \alpha,\beta>0\\
0&\text{otherwise}
\end{cases}$
$Y = \ln(X)$
$g^{-1}(y) = \exp(y)$ (as is its derivative)
$f_Y(y) = \exp(y) \cdot 
\frac{1}{\Gamma(\alpha) \beta^\alpha} [\exp(y)]^{\alpha-1} e^{-\exp(y)/\beta}$
$= \frac{1}{\Gamma(\alpha) \beta^\alpha} e^{\alpha y-\exp(y-\ln\beta)};\quad \alpha,\beta>0$
it's the negative of a Gumbel and it's of the same form as a Gompertz but not truncated to non-negative values. The $\alpha$ now acts like a scale and $\ln \beta$ as a location.
The practical questions (like "how much difference might it make to the results") were mostly explored with simulation, but the direct question of what that distribution was is readily answered using the same approach that produced that formula.
