Partial Derivatives and Gradients

Learn about partial derivatives and how to calculate the gradient of a multivariate function.

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)=πx2yf(x, y) = \pi x^2y is a multivariate function because it takes two inputs: the radius xx and the height yy of the cylinder.

For multivariate functions f(x)=f(x1,x2,...,xm)f(x) = f(x_1, x_2, ..., x_m)where the input xRmx \in \R^m is an mm-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 ff with respect to a variable xix_i (for example, radius) is obtained by varying only that variable and keeping others (such as height) constant. In other words,

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