Directional derivative

Definition at a point

For a function of two variables

Suppose $f$ is a function of two variables $x,y$. Suppose $\langle u,v\rangle$ 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 $\langle u,v\rangle$ as follows.

Item	Value
Notation	$D_{\langle u,v\rangle }(f)(x_{0},y_{0})$ or $\nabla _{\langle u,v\rangle }(f)(x_{0},y_{0})$
Definition as a limit	$\lim _{h\to 0}{\frac {f(x_{0}+uh,y_{0}+vh)-f(x_{0},y_{0})}{h}}$
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 $\langle u_{1},u_{2},\dots ,u_{n}\rangle$ 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 $\langle u_{1},u_{2},\dots ,u_{n}\rangle$ is defined as follows.

Item	Value
Notation	$D_{\langle u_{1},u_{2},\dots ,u_{n}\rangle }(f)(a_{1},a_{2},\dots ,a_{n})$ or $\nabla _{\langle u_{1},u_{2},\dots ,u_{n}\rangle }(f)(a_{1},a_{2},\dots ,a_{n})$
Definition as a limit	$\lim _{h\to 0}{\frac {f(a_{1}+u_{1}h,a_{2}+u_{2}h,\dots ,a_{n}+u_{n}h)-f(a_{1},a_{2},\dots ,a_{n})}{h}}$
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 ${\overline {x}}=\langle x_{1},x_{2},\dots ,x_{n}\rangle$. 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	$D_{\overline {u}}(f)({\overline {a}})$ or $\nabla _{\overline {u}}(f)({\overline {a}})$
Definition as a limit	$\lim _{h\to 0}{\frac {f({\overline {a}}+h{\overline {u}})-f({\overline {a}})}{h}}$
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 $\langle u,v\rangle$ is a unit vector (i.e., we have $u^{2}+v^{2}=1$). Then, the directional derivative in the direction of $\langle u,v\rangle$ 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 $\langle u,v\rangle$ at the point.

Item	Value
Notation	$D_{\langle u,v\rangle }(f)(x,y)$ or $\nabla _{\langle u,v\rangle }(f)(x,y)$
Definition as a limit	$\lim _{h\to 0}{\frac {f(x+uh,y+vh)-f(x,y)}{h}}$
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 $\langle u_{1},u_{2},\dots ,u_{n}\rangle$ 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	$D_{\langle u_{1},u_{2},\dots ,u_{n}\rangle }(f)(x_{1},x_{2},\dots ,x_{n})$ or $\nabla _{\langle u_{1},u_{2},\dots ,u_{n}\rangle }(f)(x_{1},x_{2},\dots ,x_{n})$
Definition as a limit	$\lim _{h\to 0}{\frac {f(x_{1}+u_{1}h,x_{2}+u_{2}h,\dots ,x_{n}+u_{n}h)-f(x_{1},x_{2},\dots ,x_{n})}{h}}$
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 ${\overline {x}}=\langle x_{1},x_{2},\dots ,x_{n}\rangle$. 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 $\nabla _{\overline {u}}(f)({\overline {x}})$
Definition as a limit	$\lim _{h\to 0}{\frac {f({\overline {x}}+h{\overline {u}})-f({\overline {x}})}{h}}$
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 ${\overline {x}}=\langle x_{1},x_{2},\dots ,x_{n}\rangle$. 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 $\nabla f({\overline {a}})$. Then, we have the following relationship: $D_{\overline {u}}(f)({\overline {a}})={\overline {u}}\cdot (\nabla f({\overline {a}}))$ 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 ${\overline {x}}=\langle x_{1},x_{2},\dots ,x_{n}\rangle$. Suppose ${\overline {u}}$ is a unit vector. We then have: $D_{\overline {u}}(f)({\overline {x}})={\overline {u}}\cdot (\nabla f({\overline {x}}))$ 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 ${\overline {x}}=\langle x_{1},x_{2},\dots ,x_{n}\rangle$. Suppose ${\overline {u}}$ is a unit vector. We then have: $D_{\overline {u}}(f)={\overline {u}}\cdot (\nabla f)$ 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:

$D_{\langle u_{1},u_{2},\dots ,u_{n}\rangle }(f)(x_{1},x_{2},\dots ,x_{n})=\sum _{i=1}^{n}u_{i}f_{x_{i}}(x_{1},x_{2},\dots ,x_{n})=u_{1}f_{x_{1}}(x_{1},x_{2},\dots ,x_{n})+u_{2}f_{x_{2}}(x_{1},x_{2},\dots ,x_{n})+\dots +u_{n}f_{x_{n}}(x_{1},x_{2},\dots ,x_{n})$