Let $Y$ be an exponential random variable with parameter $\frac12$. Let $X=e^{-Y/2}$. Determine the pdf of $X$.
f(t)=\frac{d}{dt}P(X\le t)=\frac{d}{dt}P(e^{-Y/2}\le t)\\=\frac{d}{dt}P(Y\ge-2\ln t)=\frac{d}{dt}(1-P(Y<-2\ln t))=-\frac12e^{\ln t}=-\frac{t}2
How come I end up with a negative value? What's wrong?