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One of the best ways to think about partial derivatives is by slicing the graph of a multivariable function. Created by Grant Sanderson.

Neither one of these derivatives tells the full story of how our function f (x, y) changes when its input changes slightly, so we call them partial derivatives.

Given a simple 2D scalar field, what is the partial derivative with respect to x or y?

Here we go over many different ways to extend the idea of a derivative to higher dimensions, including partial derivatives , directional derivatives, the gradient, vector derivatives, …

Partial derivatives tell you how a multivariable function changes as you tweak just one of the variables in its input.

The partial derivative symbol (∂) is used in multivariable calculus to indicate that you are taking a derivative with respect to one variable while keeping other variables constant.

How does the value of a multivariable function change as you nudge the input in a specific direction?

The second partial derivative test is based on a formula which seems to come out of nowhere. Here, you can see a little more intuition for why it looks the way it does.

Neither one of these derivatives tells the full story of how our function f (x, y) changes when its input changes slightly, so we call them partial derivatives.