Final answer:
The chain rule for partial derivatives is used to calculate the derivative of a function with several variables that are functions of other variables. It is done by multiplying the partial derivatives of the inner functions with the partial derivatives of the outer function with respect to the inner functions' variables.
Step-by-step explanation:
The chain rule is a formula used to compute the derivative of a composite function. In the context of partial derivatives, when you have a function of several variables that are themselves functions of other variables, the chain rule can be used to find the rate of change of the function with respect to those other variables.
For example, let's consider a function f which is a function of u and v, where u and v are both functions of x and y. The partial derivative of f with respect to x is given by:
fx = fuux + fvvx
Similarly, the partial derivative of f with respect to y would be:
fy = fuuy + fvvy
This means that to calculate the partial derivative of the function with respect to either x or y, you need to take the partial derivatives of f with respect to u and v, and multiply each by the partial derivative of u or v with respect to x or y, respectively.
The chain rule is particularly useful in multi-variable calculus and in many applications like physics, engineering, and economics where functions depend on other functions that are changing.