Extending the Derivative
We have seen how to differentiate functions of one variable, but what about functions of multiple variables? We've already seen that functions can have multiple inputs and multiple outputs, and they appear frequently in the real world. In this section, we will extend the concept of the derivative to functions of multiple variables.
📄️ The Partial Derivative
The Partial Derivative is a way to think about the derivative of a function of multiple variables.
📄️ Second Partial Derivatives
Just like how we can take second derivatives of functions of one variable, we can also take second derivatives of functions of two variables.
📄️ Gradient
The gradient of a function has multiple interpretations and uses in mathematics.
📄️ The Directional Derivative
The directional derivative is somewhat of an extension or generalization of the partial derivative.
📄️ Differentiating Vector-Valued Functions
So far, we've been dealing with functions that output a single value.
📄️ Multivariable Chain Rule
The chain rule is a fundamental concept in calculus, and it can be extended to multivariable functions.
📄️ Gradient of a Function in Cylindrical Coordinates
The gradient in cylindrical coordinates is defined a bit differently than in Cartesian coordinates.
📄️ Partial Derivatives in Cylindrical Coordinate Systems
Previously, we discussed the partial derivative of a function in Cartesian coordinates.