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 $x,y$. $(fg)_{x}=f_{x}g+fg_{x}$ $(fg)_{y}=f_{y}g+fg_{y}$ These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).

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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): $\!\nabla _{\overline {u}}(fg)=\nabla _{\overline {u}}(f)g+f\nabla _{\overline {u}}(g)$.

Statement for gradient vectors

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

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Statement for directional derivatives

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Statement for gradient vectors

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