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 $\nabla_{⟨ u, v ⟩} (f) (x_{0}, y_{0})$
Definition as a limit	$lim_{h \to 0} \frac{f (x_{0} + u h, y_{0} + v h) - f (x_{0}, y_{0})}{h}$
Definition as an ordinary derivative	$\frac{d}{d h} [f (x_{0} + u h, y_{0} + v h)] \|_{h = 0}$

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 $\nabla_{⟨ u_{1}, u_{2}, \dots, u_{n} ⟩} (f) (a_{1}, a_{2}, \dots, a_{n})$
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	$\frac{d}{d h} [f (a_{1} + u_{1} h, a_{2} + u_{2} h, \dots, a_{n} + u_{n} h)] \|_{h = 0}$

For a function of multiple variables in vector notation

Suppose $f$ is a function of a vector variable $\bar{x} = ⟨ x_{1}, x_{2}, \dots, x_{n} ⟩$ . Suppose $\bar{u}$ is a unit vector and $\bar{a}$ is a point in the domain of $f$ . The directional derivative of $f$ at $\bar{a}$ in the direction of $\bar{u}$ is denoted and defined as below.

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

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 $R^{2}$ . 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 $\nabla_{⟨ u, v ⟩} (f) (x, y)$
Definition as a limit	$lim_{h \to 0} \frac{f (x + u h, y + v h) - f (x, y)}{h}$
Definition as a partial derivative	$\frac{\partial}{p a r t i a l h} [f (x + u h, y + v h)] \|_{h = 0}$ . 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 $\nabla_{⟨ u_{1}, u_{2}, \dots, u_{n} ⟩} (f) (x_{1}, x_{2}, \dots, x_{n})$
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	$\frac{d}{d h} [f (x_{1} + u_{1} h, x_{2} + u_{2} h, \dots, x_{n} + u_{n} h)] \|_{h = 0}$ . 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 $\bar{x} = ⟨ x_{1}, x_{2}, \dots, x_{n} ⟩$ . Suppose $\bar{u}$ 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 $\nabla_{\bar{u}} (f) (\bar{x})$
Definition as a limit	$lim_{h \to 0} \frac{f (\bar{x} + h \bar{u}) - f (\bar{x})}{h}$
Definition as an ordinary derivative	$\frac{\partial}{\partial h} [f (\bar{x} + h \bar{u})] \|_{h = 0}$ . Note that we need to take the partial derivative instead of the ordinary derivative because the coordinates of $\bar{x}$ 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 $\bar{x} = ⟨ x_{1}, x_{2}, \dots, x_{n} ⟩$ . Suppose $\bar{u}$ is a unit vector and $\bar{a}$ is a point in the domain of $f$ . Suppose that the gradient vector of $f$ at $\bar{a}$ exists. We denote this gradient vector by $\nabla f (\bar{a})$ . Then, we have the following relationship: $D_{\bar{u}} (f) (\bar{a}) = \bar{u} \cdot (\nabla f (\bar{a}))$ 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 $\bar{x} = ⟨ x_{1}, x_{2}, \dots, x_{n} ⟩$ . Suppose $\bar{u}$ is a unit vector. We then have: $D_{\bar{u}} (f) (\bar{x}) = \bar{u} \cdot (\nabla f (\bar{x}))$ 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 $\bar{x} = ⟨ x_{1}, x_{2}, \dots, x_{n} ⟩$ . Suppose $\bar{u}$ is a unit vector. We then have: $D_{\bar{u}} (f) = \bar{u} \cdot (\nabla f)$ The right side here is a dot product of vector-valued functions (the constant function $\bar{u}$ 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:

$D_{⟨ u_{1}, u_{2}, \dots, u_{n} ⟩} (f) (x_{1}, x_{2}, \dots, x_{n}) = \sum_{i = 1}^{n} u_{i} f_{x_{i}} (x_{1}, x_{2}, \dots, x_{n}) = u_{1} f_{x_{1}} (x_{1}, x_{2}, \dots, x_{n}) + u_{2} f_{x_{2}} (x_{1}, x_{2}, \dots, x_{n}) + \dots + u_{n} f_{x_{n}} (x_{1}, x_{2}, \dots, x_{n})$