Product rule for partial differentiation

Statement for two functions

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. 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.
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.

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: $\!\nabla _{\overline {u}}(fg)({\overline {x_{0}}})=f({\overline {x_{0}}})\nabla _{\overline {u}}(g)({\overline {x_{0}}})+g({\overline {x_{0}}})\nabla _{\overline {u}}(f)({\overline {x_{0}}})$
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: $\!\nabla _{\overline {u}}(fg)({\overline {x}})=f({\overline {x}})\nabla _{\overline {u}}(g)({\overline {x}})+g({\overline {x}})\nabla _{\overline {u}}(f)({\overline {x}})$ .
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: $\!\nabla _{\overline {u}}(fg)=f\nabla _{\overline {u}}(g)+g\nabla _{\overline {u}}(f)$ .

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

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: $\!\nabla (fg)({\overline {x_{0}}})=f({\overline {x_{0}}})\nabla (g)({\overline {x_{0}}})+g({\overline {x_{0}}})\nabla (f)({\overline {x_{0}}})$ . 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: $\!\nabla _{\overline {u}}(fg)=f({\overline {x}})\nabla (g)({\overline {x}})+g({\overline {x}})\nabla (f)({\overline {x}})$ . Note that the products on the right side are scalar-vector 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: $\!\nabla (fg)=f\nabla (g)+g\nabla (f)$ . Note that the products on the right side are scalar-vector multiplications.

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