# Difference between revisions of "Product rule for partial differentiation"

## Statement for two functions

### Statement for partial derivatives for functions of two variables

The derivatives used here are partial derivatives.

Version type Statement
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. Let $fg$ denote the product of the functions. Then, we have:
$(fg)_x(x_0,y_0) =f_x(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_x(x_0,y_0)$
Suppose the partial derivatives $f_y(x_0,y_0)$ and $g_y(x_0,y_0)$ both exist. Then, we have:
$(fg)_y(x_0,y_0) = f_y(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_y(x_0,y_0)$
generic point, named functions Suppose $f,g$ are both functions of variables $x,y$.
$(fg)_x(x,y) =f_x(x,y)g(x,y) + f(x,y)g_x(x,y)$
$(fg)_y(x,y) = f_y(x,y)g(x,y) + f(x,y)g_y(x,y)$
These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).
generic point, named functions, point-free notation Suppose $f,g$ are both functions of variables $x,y$.
$(f g)_x =f_xg + fg_x$
$(f g)_y = f_yg + fg_y$
These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).

### Statement for partial derivatives for functions of multiple variables

Version type Statement
specific point, named functions Suppose $f,g$ are both functions of variables $x_1,x_2,\dots,x_n$. Suppose $(a_1,a_2,\dots,a_n)$ is a point in the domain of both $f$ and $g$. Fix a number $i$ in $\{ 1,2,3,\dots,n \}$. Suppose the partial derivatives $f_{x_i}(a_1,a_2,\dots,a_n)$ and $g_{x_i}(a_1,a_2,\dots,a_n)$ both exist. Let $fg$ denote the product of the functions. Then, we have:
$(fg)_{x_i}(a_1,a_2,\dots,a_n) =f_{x_i}(a_1,a_2,\dots,a_n)g(a_1,a_2,\dots,a_n) + f(a_1,a_2,\dots,a_n)g_{x_i}(a_1,a_2,\dots,a_n)$
generic point, named functions Suppose $f,g$ are both functions of variables $x_1,_2,\dots,x_n$. Then, for any fixed $i$ in $\{ 1,2,3,\dots,n \}$: $(fg)_{x_i}(x_1,x_2,\dots,x_n) =f_{x_i}(x_1,x_2,\dots,x_n)g(x_1,x_2,\dots,x_n) + f(x_1,x_2,\dots,x_n)g_{x_i}(x_1,x_2,\dots,x_n)$
These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).
generic point, named functions, point-free notation Suppose $f,g$ are both functions of variables $x,y$. Then, for any fixed $i$ in $\{ 1,2,3,\dots,n \}$: $(fg)_{x_i} = f_{x_i}g + fg_{x_i}$
These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).

### Statement for directional derivatives

Version type Statement
specific point, named functions Suppose $f,g$ are both real-valued functions of a vector variable $\overline{x}$. Suppose $\overline{u}$ is a unit vector. Suppose $\overline{x_0}$ is a point in the domain of both functions. Then, we have the following product rule for directional derivatives:
$\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = \nabla_{\overline{u}}(f)(\overline{x_0})g(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})$
generic point, named functions Suppose $f,g$ are both real-valued functions of a vector variable $\overline{x}$. Suppose $\overline{u}$ is a unit vector. Then, we have the following product rule for directional derivatives wherever the right side expression makes sense (see concept of equality conditional to existence of one side):
$\! \nabla_{\overline{u}}(fg)(\overline{x}) = \nabla_{\overline{u}}(f)(\overline{x})g(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})$.
generic point, named functions, point-free notation Suppose $f,g$ are both real-valued functions of a vector variable $\overline{x}$. Suppose $\overline{u}$ is a unit vector. Then, we have the following product rule for directional derivatives wherever the right side expression makes sense (see concept of equality conditional to existence of one side):
$\! \nabla_{\overline{u}}(fg) = \nabla_{\overline{u}}(f)g + f\nabla_{\overline{u}}(g)$.

Version type Statement
specific point, named functions Suppose $f,g$ are both real-valued functions of a vector variable $\overline{x}$. Suppose $\overline{x_0}$ is a point in the domain of both functions. Then, we have the following product rule for gradient vectors:
$\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0})$. Note that the products on the right side are scalar-vector multiplications.
generic point, named functions Suppose $f,g$ are both real-valued functions of a vector variable $\overline{x}$. Then, we have the following product rule for gradient vectors wherever the right side expression makes sense (see concept of equality conditional to existence of one side):
$\! \nabla(fg)(\overline{x})= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})$. Note that the products on the right side are scalar-vector function multiplications.
generic point, named functions, point-free notation Suppose $f,g$ are both real-valued functions of a vector variable $\overline{x}$. Then, we have the following product rule for gradient vectors wherever the right side expression makes sense (see concept of equality conditional to existence of one side):
$\! \nabla(fg) = g\nabla (f) + f\nabla (g)$. Note that the products on the right side are scalar-vector function multiplications.

## Statement for multiple functions

### Statement for partial derivatives

Fill this in later

### Statement for directional derivatives

Fill this in later