Why do you use gradient?

Updated: 10/18/2022
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The derivative of a function, df/dx, is to single variable calculus as the gradient of a function, ∇f, is to multivariable calculus.

If f is a function of three variables, x, y, and z, then the gradient of f is the vector function ∇f(x, y, z) = <∂f/∂x(x, y, z), ∂f/∂y(x, y, z), ∂f/∂z(x, y, z)>

All of the uses of derivatives in single variable calculus are analogous to the uses of gradients in multivariable calculus:

In single variable calculus the derivative tells us the instantaneous rate of change at some point, [x, f(x)]. In multivariable calculus, the gradient of a function tells us the instantaneous rate of change at some point, [x, y, f(x,y)], or if the function is of more than two variables, ∇f would tell us the instantaneous rate of change at a point [x, y, z, ….., f(x, y, z, ….)]. One Important difference in calculus of more than one variable is that a function can have many different rates of changes at one point. To understand why this is so, imagine that you are standing on a hilltop which is defined by a function of two variables f(x, y). The downward slope of the hill, the gradient, is different depending on the direction you look; to find the slope you need to specify a direction. This is why we take the 'directional derivative' which is simply the dot product of the gradient with a unit direction vector (the direction you are looking down the hill). For example suppose we want to find the instantaneous rate of change of the function f(x,y) = x2 + y2 at the point (2,1) in the direction of v = <0, 1>:

The directional derivative in the direction of v = ∇f(x, y) ● <0,1>

= < ∂f/∂x(x, y, z), ∂f/∂y(x, y, z)> ● <0,1> = <2x, 2y> ● <0,1> = 2y evaluated at (2,1) = 2.

Let's continue our comparison of derivatives and gradients. In single variable calculus a derivative of a function is equal to zero at a maximum or minimum value of the function. This fact can be used in practical applications that require maximizing and minimizing functions of one variable. The same is said of the gradient in multivariable calculus. By setting the gradient of a multivariable function equal to zero, we can solve for the point of maximum or minimum values.

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