I am not very familiar with multivariable calculus, but something tells me that I don't need to be in order to solve this problem; take a look:
Suppose that $X_1,...,X_m$ and $Y_1,...,Y_n$ are independent exponential random variables with $X_i\sim EXP(\lambda)$ and $Y_j\sim EXP(\theta \lambda)$.
Find the $MLE$ of $\lambda$ and $\theta$.
Finding the MLE of $\lambda$ is simple; by ignoring the $Y_j$ altogether and just looking at the $X_i$, it turns out to be $\sum x_i/m$. However, for $\theta$, I am no longer sure since the distribution of $Y_j$ is also dependent on $\lambda$. I don't know if I need to go as far as finding the gradient or if I can somehow use my previous result, but either way, I honestly don't know how to do it.
Any advice would be appreciated.