Examples

Partial derivatives of a monomial

Partial derivatives of a monomial

Problem: Compute the partial derivatives of first and second order of the function $f(x,y)=x^3{y}^5$.
Solution: We have $f_x=3x^2{y}^5$, $\frac{\partial f}{\partial y}=5x^3{y}^4$, $f_{xy}=f_{yx}=15x^2{y}^4$, $\frac{{\partial }^{2}f}{\partial x^2}=6x{y}^5$, $f_{yy}=20x^3{y}^3$.