Final answer:
The subject of this question is the Partial Derivative Chain Rule in Mathematics. The chain rule is used to find the rate of change of a dependent variable with respect to an independent variable in multivariable calculus. To use the chain rule for partial derivatives, we differentiate each component of the function one at a time and then multiply the derivatives according to the chain rule formula.
Step-by-step explanation:
The subject of this question is the Partial Derivative Chain Rule in Mathematics. The chain rule is a formula for taking the derivative of a composite function. It is used to find the rate of change of a dependent variable with respect to an independent variable in multivariable calculus.
To use the chain rule for partial derivatives, we differentiate each component of the function one at a time. Let's say we have a function z = f(x, y) and we want to find ∂z/∂x. We differentiate f with respect to x, while treating y as a constant. We then multiply it by ∂x/∂x and sum it with ∂y/∂x multiplied by ∂z/∂y.
For example, if we have z = x^2y + y^2 and we want to find ∂z/∂x, we differentiate x^2y with respect to x, which gives us 2xy. We then differentiate y^2 with respect to x, treating y as a constant, which gives us 0. So the partial derivative of x^2y with respect to x is 2xy. The partial derivative of y^2 with respect to x is 0. Therefore, ∂z/∂x = 2xy + 0 = 2xy.