00:05
We're given a function and we're asked to use the definition of partial derivatives as limits to find the partial derivatives of this function.
00:12
The function is f of x y equals x over x plus y squared.
00:23
Now by definition, the partial derivative of f with respect to x is the limit as h approaches 0 of f of x plus h y minus f of x y over h, which is the limit as h approaches 0 of x plus h divided by x plus h plus y squared minus x over x plus y squared.
01:10
All over h, this is the limit as h approaches zero.
01:19
Of, well, i'm going to combine some terms here.
01:28
You have x plus h.
02:00
This is x plus h times x plus y squared minus x times x plus h plus y squared, all over an h times an x plus h plus y squared, times x plus y squared.
02:26
This is the limit as each approach is zero of, well, we have x squared plus xy squared plus hx plus hy squared minus x squared minus h x squared minus x y squared, so we can simplify our numerator by canceling out the x squared terms and the xy squared terms.
03:03
So we just have an h all over h times x plus h plus y squared times x plus y squared...