Product rule for partial differentiation

Statement for two functions

Statement for partial derivatives

Version type	Statement for functions of two variables
specific point, named functions	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: $(f\cdot g{)}_x(x_0,y_0)=f_x(x_0,y_0)g(x_0,y_0)+f(x_0,y_0)g_x(x_0,y_0)$ Suppose the partial derivatives $f_y(x_0,y_0)$ and $g_y(x_0,y_0)$ both exist. Then, we have: $(f\cdot g{)}_y(x_0,y_0)=f_y(x_0,y_0)g(x_0,y_0)+f(x_0,y_0)g_y(x_0,y_0)$
generic point, named functions	Suppose $f,g$ are both functions of variables $x,y$. $(f\cdot g{)}_x(x,y)=f_x(x,y)g(x,y)+f(x,y)g_x(x,y)$ $(f\cdot g{)}_y(x,y)=f_y(x,y)g(x,y)+f(x,y)g_y(x,y)$ These hold wherever the right side expressions make sense.
generic point, named functions, point-free notation	Suppose $f,g$ are both functions of variables $x,y$. $(f\cdot g{)}_x=f_x\cdot g+f\cdot g_x$ $(f\cdot g{)}_y=f_y\cdot g+f\cdot g_y$ These hold wherever the right side expressions make sense.

Statement for directional derivatives

Suppose $f,g$ are both real-valued functions of many variables. Suppose $\overline{u}$ is a unit vector. Then, we have the following product rule for directional derivatives:

${\mathrm{\nabla }}_{\overline{u}}(f\cdot g){|}_{\overline{x}}=f{\mathrm{\nabla }}_{\overline{u}}(g)+g{\mathrm{\nabla }}_{\overline{u}}(f)$

The rule applies at all points where the right side make sense.

Statement for gradient vectors

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Statement for multiple functions

Statement for partial derivatives

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Statement for directional derivatives

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Statement for gradient vectors

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