Applications of multivariable derivatives

The tools of partial derivatives, the gradient, etc. can be used to optimize and approximate multivariable functions. These are very useful in practice, and to a large extent this is why people study multivariable calculus.

How to compute the tangent plane of a three-dimensional graph, and how this can be generalized to approximations of higher dimensional functions.

In practice, people don't play with multivariable functions just for fun. Quite often, you want to optimize something, like the profit of a company or the air resistance of a plane. Using the tools of multivariable derivatives, we can find the maximum or minimum values of many multivariable functions.

How do you find the maximum or minimum of a multivariable function if you are constrained to only look at certain inputs? For example, maybe you want to maximize your companies profits, but you have certain budgetary constraints.