mathematics calculus

Question

Calculate ∂f / ∂x and ∂f /∂y if: a) f(x,y) = 3x³y - lny b) f(x,y) = xe^y / 1+x²y and please explain step by step!

Answer 1

I will do the first one for you; you should (with my direction) be able to do the 2nd.

For a)

partial f/with respect to x = 9x²y and partial f/with respect to y = 3x³-(1/y).

To compute the partial derivative with respect to x, you simply consider y to be a constant and compute the derivative with respect to x. Similarly to compute the partial with respect to y, you simply treat x as a constant.

Here is the partial with respect to x.

You mimic this to get the partial with respect to y.

[(1+x²y)e^y-xe^y(2xy)]/(1+x²y)²

To get the partial derivative with respect to y, you think of x as a constant and differentiate with respect to y.