Chain Rule in Calculus

The Chain Rule in calculus is a fundamental rule for finding the derivative of a composition of two functions. If you have two functions, f(x) and g(x), and you want to find the derivative of their composition, f(g(x)), the chain rule states that:

f'(g(x)) = f'(g(x)) × g'(x)

Explanation

– f(g(x)) is the composition of the functions f and g.

– The derivative of the composition is found by taking the derivative of the outer function f evaluated at g(x), and multiplying it by the derivative of the inner function g(x).

Example

Suppose we want to differentiate the function h(x) = sin(x²).

1. Identify the inner and outer functions:
   • The outer function is f(u) = sin(u), where u = x².
   • The inner function is g(x) = x².
2. Apply the chain rule:

   f'(x) = cos(x²) × (d/dx)(x²)

3. The derivative of x² is 2x, so:

   f'(x) = cos(x²) × 2x

Thus, the derivative of sin(x²) is 2x cos(x²).

The chain rule is essential for differentiating complex functions where one function is nested inside another. It helps us calculate derivatives of composite functions in an easy and structured way.