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The chain rule in calculus is a fundamental theorem for differentiating composite functions, such as those involving polynomials or trigonometric expressions. It breaks down complex functions into simpler parts, making the process of finding derivatives more manageable. This rule is also pivotal for integration techniques, where it is used in reverse as the substitution method. Mastery of the chain rule is crucial for advanced calculus studies and applications.

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## Definition of the Chain Rule

### Composite Functions

The chain rule is used to find the derivative of composite functions, which are functions made up of other functions

### Derivative of Composite Functions

Product of Derivatives

The chain rule states that the derivative of a composite function is the product of the derivatives of its component functions

Leibniz's Notation

In Leibniz's notation, the chain rule is expressed as the product of the derivatives of the inner and outer functions

### Applications of the Chain Rule

The chain rule is crucial for differentiating a wide variety of functions, including polynomial, trigonometric, and implicitly defined functions

## The Reverse Chain Rule in Integration

### Method of Substitution

The reverse chain rule, also known as the method of substitution, is used in integration to reverse the process of differentiation

### Example of Integration

The reverse chain rule can be used to integrate functions by substituting a variable and then integrating the resulting expression