Directional derivative

Definition at a point

For a function of two variables

Suppose $f$ is a function of two variables $x,y$. Suppose $⟨u,v⟩$ is a unit vector (i.e., we have $u^2+v^2=1$). Suppose $(x_0,y_0)$ is a point in the domain of $f$ We define the directional derivative of $f$ at $(x_0,y_0)$ in the direction of $⟨u,v⟩$ as follows.

Item	Value
Notation	$D_{⟨u,v⟩}(f)(x_0,y_0)$ or ${\mathrm{\nabla }}_{⟨u,v⟩}(f)(x_0,y_0)$
Definition as a limit	$\underset{h\to 0}{lim}\frac{f(x_0+uh,y_0+vh)-f(x_0,y_0)}{h}$
Definition as an ordinary derivative	$\frac{d}{dh}[f(x_0+uh,y_0+vh)]{|}_{h=0}$

For a function of multiple variables

Suppose $f$ is a function of variables $x_1,x_2,\dots ,x_n$. Suppose $⟨u_1,u_2,\dots ,u_n⟩$ is a unit vector (i.e., we have $u_1^2+u_2^2+\dots +u_n^2=1$). Suppose $(a_1,a_2,\dots ,a_n)$ is a point in the domain of $f$. The directional derivative of $f$ at $(a_1,a_2,\dots ,a_n)$ in the direction of $⟨u_1,u_2,\dots ,u_n⟩$ is defined as follows.

Item	Value
Notation	$D_{⟨u_1,u_2,\dots ,u_n⟩}(f)(a_1,a_2,\dots ,a_n)$ or ${\mathrm{\nabla }}_{⟨u_1,u_2,\dots ,u_n⟩}(f)(a_1,a_2,\dots ,a_n)$
Definition as a limit	$\underset{h\to 0}{lim}\frac{f(a_1+u_{1}h,a_2+u_{2}h,\dots ,a_n+u_{n}h)-f(a_1,a_2,\dots ,a_n)}{h}$
Definition as an ordinary derivative	$\frac{d}{dh}[f(a_1+u_{1}h,a_2+u_{2}h,\dots ,a_n+u_{n}h)]{|}_{h=0}$

For a function of multiple variables in vector notation

Suppose $f$ is a function of a vector variable $\overline{x}=⟨x_1,x_2,\dots ,x_n⟩$. Suppose $\overline{u}$ is a unit vector and $\overline{a}$ is a point in the domain of $f$. The directional derivative of $f$ at $\overline{a}$ in the direction of $\overline{u}$ is denoted and defined as below.

Item	Value
Notation	Failed to parse (syntax error): {\displaystyle D_{\overline{u}(f)(\overline{a})} or ${\mathrm{\nabla }}_{\overline{u}}(f)(\overline{a})$
Definition as a limit	$\underset{h\to 0}{lim}\frac{f(\overline{a}+h\overline{u})-f(\overline{a})}{h}$
Definition as an ordinary derivative	$\frac{d}{dh}[f(\overline{a}+h\overline{u})]{|}_{h=0}$

Definition as a function

For a function of two variables

Suppose $f$ is a function of two variables $x,y$, with domain a subset of ${\mathbb{R}}^2$. Suppose $⟨u,v⟩$ is a unit vector (i.e., we have $u^2+v^2=1$). Then, the directional derivative in the direction of $⟨u,v⟩$ is a function with domain a subset of the domain of $f$, defined as the function that sends any point in the domain of $f$ to the directional derivative of $f$ in the direction of $⟨u,v⟩$ at the point.

Item	Value
Notation	$D_{⟨u,v⟩}(f)(x,y)$ or ${\mathrm{\nabla }}_{⟨u,v⟩}(f)(x,y)$
Definition as a limit	$\underset{h\to 0}{lim}\frac{f(x+uh,y+vh)-f(x,y)}{h}$
Definition as a partial derivative	$\frac{\partial }{partialh}[f(x+uh,y+vh)]{|}_{h=0}$. Note that we need to use a partial derivative because $x,y$ are now variable as we are not doing this at a single point.

For a function of multiple variables

Suppose $f$ is a function of variables $x_1,x_2,\dots ,x_n$. Suppose $⟨u_1,u_2,\dots ,u_n⟩$ is a unit vector (i.e., we have $u_1^2+u_2^2+\dots +u_n^2=1$). We define and denote the directional derivative as below.

Item	Value
Notation	$D_{⟨u_1,u_2,\dots ,u_n⟩}(f)(x_1,x_2,\dots ,x_n)$ or ${\mathrm{\nabla }}_{⟨u_1,u_2,\dots ,u_n⟩}(f)(x_1,x_2,\dots ,x_n)$
Definition as a limit	$\underset{h\to 0}{lim}\frac{f(x_1+u_{1}h,x_2+u_{2}h,\dots ,x_n+u_{n}h)-f(x_1,x_2,\dots ,x_n)}{h}$
Definition as an ordinary derivative	$\frac{d}{dh}[f(x_1+u_{1}h,x_2+u_{2}h,\dots ,x_n+u_{n}h)]{|}_{h=0}$. Note that we need to use a partial derivative because $x_1,x_2,\dots ,x_n$ are now variable as we are not doing this at a single point.

For a function of multiple variables in vector notation

Suppose $f$ is a function of a vector variable $\overline{x}=⟨x_1,x_2,\dots ,x_n⟩$. Suppose $\overline{u}$ is a unit vector. We define and denote the directional derivative of $f$ in the direction of $u$ below.

Item	Value
Notation	Failed to parse (syntax error): {\displaystyle D_{\overline{{u}}(f)(\overline{x})} or ${\mathrm{\nabla }}_{\overline{u}}(f)(\overline{x})$
Definition as a limit	$\underset{h\to 0}{lim}\frac{f(\overline{x}+h\overline{u})-f(\overline{x})}{h}$
Definition as an ordinary derivative	$\frac{\partial }{\partial h}[f(\overline{x}+h\overline{u})]{|}_{h=0}$. Note that we need to take the partial derivative instead of the ordinary derivative because the coordinates of $\overline{x}$ themselves are not fixed, as we are doing this at a generic rather than a fixed point.

Relation with gradient vector

Version type	Statement
at a point, in vector notation (multiple variables)	Suppose $f$ is a function of a vector variable $\overline{x}=⟨x_1,x_2,\dots ,x_n⟩$. Suppose $\overline{u}$ is a unit vector and $\overline{a}$ is a point in the domain of $f$. Suppose that the gradient vector of $f$ at $\overline{a}$ exists. We denote this gradient vector by $\mathrm{\nabla }f(\overline{a})$. Then, we have the following relationship: $D_{\overline{u}}(f)(\overline{a})=\overline{u}\cdot (\mathrm{\nabla }f(\overline{a}))$ The right side here is the dot product of vectors.
generic point, in vector notation (multiple variables)	Suppose $f$ is a function of a vector variable $\overline{x}=⟨x_1,x_2,\dots ,x_n⟩$. Suppose $\overline{u}$ is a unit vector. We then have: $D_{\overline{u}}(f)(\overline{x})=\overline{u}\cdot (\mathrm{\nabla }f(\overline{x}))$ The right side here is a dot product of vectors. The equality holds whenever the right side makes sense.
generic point, point-free notation (multiple variables)	Suppose $f$ is a function of a vector variable $\overline{x}=⟨x_1,x_2,\dots ,x_n⟩$. Suppose $\overline{u}$ is a unit vector. We then have: $D_{\overline{u}}(f)=\overline{u}\cdot (\mathrm{\nabla }f)$ The right side here is a dot product of vector-valued functions (the constant function $\overline{u}$ and the gradient vector of $f$). The equality holds whenever the right side makes sense.

Relation with partial derivatives

If the gradient vector at a point exists, then it is a vector whose coordinates are the corresponding partial derivatives of the function. Thus, conditional to the existence of the gradient vector, we have that:

$D_{⟨u_1,u_2,\dots ,u_n⟩}(f)(x_1,x_2,\dots ,x_n)=\sum _{i=1}^n{u}_i{f}_{x_i}(x_1,x_2,\dots ,x_n)=u_1{f}_{x_1}(x_1,x_2,\dots ,x_n)+u_2{f}_{x_2}(x_1,x_2,\dots ,x_n)+\dots +u_n{f}_{x_n}(x_1,x_2,\dots ,x_n)$