Calculus 3 Derivative

Question

Given f(x,y)=−x^3+xy^6−3y^4, find fxx(x,y) and fxy(x,y)

Answer 1

Let's work through a similar problem: f(x,y) = x^2 + x^2y^3 - 2y^4.

To find both f_xx(x,y) and f_xy(x,y), we need to first differentiate with respect to x. This is simply an application of polynomial rule, in which we treat every instance of y the way we would treat any other constant.

Then f_x(x,y) = 2x +2xy^3. Note that, for example, the 2y^4 term has disappeared, because we are differentiating with respect to x.

To compute fxx(x,y), we simply differentiate with respect to x again: fxx(x,y) = 2 2y^3.

To compute fxy(x,y), we differentiate 2x + 2xy^3 with respect to y - treating x as a constant. This gives us fxy(x,y) = 6xy^2.