I have a question about finding the MLE..

$\displaystyle Let\;\; $X_1, \cdots, X_n$ \mbox{be a random sample from a population with pdf}\;\;$f_{\theta}(x)=(\theta+1)x^{\theta}$

$\displaystyle ;\;0<x<1;\;\mbox{where }\theta>-1$ \mbox{is an unknown parameter}.$

Find the MLE for $\displaystyle e^{\theta}$

I am confused on how to start this. To find the MLE of $\displaystyle \theta$, we would find the likelihood function of $\displaystyle L(\theta|x)$. so does this mean the to find the MLE for the MLE for $\displaystyle e^{\theta}$, I would go like this:

$\displaystyle L(e^{\theta}|x)= \prod_{i=1}^n (e^{\theta}+1){x_i}^{e^{\theta}}$

is this right??