vela said:You need to use the chain rule. Don't forget z still varies with x.
Partial differentiation is a mathematical process used to find the rate of change of a multivariable function with respect to one of its variables while holding all other variables constant. It is commonly used in the field of calculus and is an essential tool in solving problems involving multiple variables.
The basic rules of partial differentiation include the product rule, quotient rule, and chain rule. The product rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. The quotient rule states that the derivative of a quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. The chain rule states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
The main difference between partial differentiation and ordinary differentiation is that in partial differentiation, we are taking the derivative with respect to one variable while holding all other variables constant. In ordinary differentiation, we are taking the derivative with respect to a single variable that is independent of other variables.
Yes, partial differentiation can be applied to any function that has multiple variables. However, the function must be continuous and differentiable for partial differentiation to be valid.
Partial differentiation has many practical applications in fields such as physics, economics, engineering, and statistics. It is used to solve optimization problems, find maximum and minimum values, and model relationships between multiple variables.