Gradient of a Function
Let be a differentiable function, then the vector is called the gradient of function , and is defined as
(1)
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 , the derivative of function in the direction of the unit vector is
(2)
Maximum Value of the Directional Derivative
We can write the derivative of a function in the direction of a unit vector as
(3)
and , so the maximum value of the derivative is
(4)
Watch this video to understand the physical interpretation of directional derivative.
Reference
Dennis G. Zill. Advanced Engineering Mathematics, edition. Jones Bartlett Learning. 2016.