Product rule for partial differentiation

Statement for two functions

Statement for partial derivatives

Version type

Statement for functions of two variables

specific point, named function

Suppose $f,g$ are both functions of variables $x,y$. Suppose $(x_0,y_0)$ is a point in the domain of both $f$ and $g$. Suppose the partial derivatives $f_x(x_0,y_0)$ and $g_x(x_0,y_0)$ both exist. Then, we have: $(f\cdot g{)}_x(x_0,y_0)=f_x(x_0,y_0)g(x_0,y_0)+f(x_0,y_0)g_x(x_0,y_0)$ uppose the partial derivatives $f_y(x_0,y_0)$ and $g_y(x_0,y_0)$ both exist. Then, we have: $(f\cdot g{)}_y(x_0,y_0)=f_y(x_0,y_0)g(x_0,y_0)+f(x_0,y_0)g_y(x_0,y_0)$

Statement for directional derivatives

Fill this in later

Statement for gradient vectors

Fill this in later

Statement for multiple functions

Statement for partial derivatives

Fill this in later

Statement for directional derivatives

Fill this in later

Statement for gradient vectors

Fill this in later