Chain Rule

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Chain Rule

The chain rule is a method of differentiating a function which is composed of two nested functions. That is, the chain rule provides a general case for taking the x derivative of a function f(g(x)). The chain rule states that the derivative of such a nested function is the product of the derivatives of each function, evaluated at appropriate points. That is, the derivative of such a function is the derivative of f, assuming that g is itself the variable, multiplied by the derivative of g, assuming that x is the variable.

For example, the function sin(x²) can be evaluated using the chain rule.

The derivative of sin(y) is cos(y), and the derivative of x² is 2x. Therefore, the derivative of sin(x²) is 2x cos(x²). The term that is particularly of interest is the argument of the cosine term. The only trick to the chain rule is remembering that the original argument of the outer function must be the argument of its derivative as well.

The chain rule is one of the basic rules of differentiation. It would be possible to step through the limit definition of a derivative every time a composite or nested function came up, but the chain rule provides a much more efficient method and is widely used wherever calculus is applied.