Partial Derivatives and Gradients

What are gradients?

A multivariate function in mathematics is a function that takes more than one input variable and produces one or more output variables. For example, the volume of a cylinder $f(x, y) = \pi x^2y$ is a multivariate function because it takes two inputs: the radius $x$ and the height $y$ of the cylinder.

For multivariate functions $f(x) = f(x_1, x_2, ..., x_m)$where the input $x \in \R^m$ is an $m$-dimensional vector, the generalization of the derivative is known as the gradient. The gradient is a collection of quantities known as partial derivatives.

Partial derivatives are useful for multivariate functions, such as the volume of a cylinder, because they can help us understand how the volume changes when one of the variables (radius or height) changes while the other is fixed. 

The partial gradient of the function $f$ with respect to a variable $x_i$ (for example, radius) is obtained by varying only that variable and keeping others (such as height) constant. In other words,