Directional Derivative

Gradient of a Function

Let f(x,y,z) be a differentiable function, then the vector \boldsymbol{\nabla}f is called the gradient of function f, and is defined as

(1)   \begin{equation*}\nabla f(x,y,z) = \frac{\partial f}{\partial x} \boldsymbol{i} + \frac{\partial f}{\partial y} \boldsymbol{j} + \frac{\partial f}{\partial z} \boldsymbol{k} .\end{equation*}


Gradient of a function at any location, gives us the direction that we can move to reach the maximum of that function. However, while moving in that direction, we still need to check the gradient at different locations to reach the maximum.
Watch this video to understand the physical interpretation of gradients.

Directional Derivative

By defining a unit vector \boldsymbol{u} = \cos \theta \boldsymbol{i} + \sin \theta \boldsymbol{j}, the derivative of function f in the direction of the unit vector \boldsymbol{u} is

(2)   \begin{equation*}D_u f(x,y,z) = \boldsymbol{\nabla} f(x,y,z) . \boldsymbol{u} .\end{equation*}

Maximum Value of the Directional Derivative

We can write the derivative of a function f(x,y,z) in the direction of a unit vector \boldsymbol{u} as

(3)   \begin{equation*}D_u f = || \boldsymbol{\nabla} f || ~ || \boldsymbol{u} || \cos \phi = || \boldsymbol{\nabla} f || \cos \phi ,\end{equation*}


and -1 \leq \cos \phi \leq 1, so the maximum value of the derivative is

(4)   \begin{equation*}D_u f = || \boldsymbol{\nabla} f || .\end{equation*}


Watch this video to understand the physical interpretation of directional derivative.

Reference

Dennis G. Zill. Advanced Engineering Mathematics, 6^{th} edition. Jones & Bartlett Learning. 2016.