# How do you take the partial derivative of this monster?

$$e^{10x -x^2 +4y -y^2}$$

I don't know where to start. I have a gut feeling this might require the chain rule, but I don't know how to use it on this thing. I tried doing some silly simplification which resulted in a pair of quotients and products of exponentials and tried to derive those using the quotient rule but it didn't work.

In any case, I need to find the first and second partial derivatives. How do I go about finding them?

Hey Raziel,

Try rewriting a more general (and less messy expression) like:

$$e^{10x -x^2 +4y -y^2} = e^{f(x,y)}$$

Taking the partial derivative (w.l.o.g. with respect to x) of the above expression is just taking the derivative of the expression with respect to x while holding y constant. In other words, what is,

$$\frac{d}{dx}e^{f(x)}\mathrm{?}$$

If you know this piece of information, then you should be able to evaluate,

$$\frac{\partial}{\partial x}e^{f(x,y)},$$

since they follow the same evaluation process.

Got it, thanks.