# Difference between revisions of "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. 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 $x,y$. $(f g)_x =f_xg + fg_x$ $(f g)_y = f_yg + fg_y$
These hold wherever the right side expressions make sense.

### 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: $\! \nabla_{\overline{u}}(fg)(\overline{x}) = + \nabla_{\overline{u}}(f)(\overline{x})g(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})$.
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: $\! \nabla_{\overline{u}}(fg) = \nabla_{\overline{u}}(f)g + f\nabla_{\overline{u}}(g)$.

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.

## Statement for multiple functions

### Statement for partial derivatives

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### Statement for directional derivatives

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