Understanding the Chain Rule | Simplifying the Process of Finding Derivatives with Composite Functions

Chain Rule (Using ‘ )

The chain rule is a fundamental rule in calculus that enables us to find the derivative of a composite function

The chain rule is a fundamental rule in calculus that enables us to find the derivative of a composite function. In other words, if we have a function that is formed by applying one function to the output of another function, the chain rule allows us to find the derivative of the composite function by breaking it down into derivatives of the individual functions involved.

The chain rule can be stated as follows: Let’s say we have two functions, f(x) and g(x), such that we want to find the derivative of the composite function h(x) = f(g(x)). The chain rule states that the derivative of h(x) with respect to x can be found by multiplying the derivative of the outer function f(g(x)) with respect to its input (g'(x)), by the derivative of the inner function g(x) with respect to its input (f'(g(x))). Mathematically, it can be written as:

(h(x))’ = (f(g(x)) )’ = f'(g(x)) * g'(x)

The prime notation (‘) is used to indicate the derivative with respect to x. So, (h(x))’ represents the derivative of h(x) with respect to x, f'(g(x)) represents the derivative of f(g(x)) with respect to g(x), and g'(x) represents the derivative of g(x) with respect to x.

The chain rule is particularly useful when dealing with complex functions composed of several nested functions. By applying the chain rule repeatedly, we can find the derivative of the composite function step by step, simplifying the process of finding derivatives.

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