Calc 3 Easy Questions: Partial Derivative of g(x,y)

[don23]

calc 3 easy question!

I am trying to find the partial derivative of the following function with respect to y. I know how to find it without using the definition...but i want to know how to do it both ways. any help??

g(x,y)=x^2*e^-y

I got: lim as h approaches h [(x^2*e^-(y+h))-x^2*e^-y]/h...simplifying it is the hard part. ...any step by step help would be greatly appreciated.

 

[SN1987a]

I think you need to use l'Hospital's rule for limits (heard of it?). Anyways, from the point you got to, you can factor x^2*e^(-y) from the expression, and since the only variable with respect to the limit is h, you can take x^2*e^(-y) out of the limit.

Then you're left with lim(e^(-h)-1)/h, which is indeterminate (0/0). Here's where you can use l'Hospital's rule, which states that for an indeterminate ratio of two expressions, the limit of the ratio is equal to the limit of the ratio of the derivatives. Once you do this, the limit is quite friendly, and you can see it is actually equal to -1.

Helpful?