Chain Rule

Understanding the Chain Rule

• The Chain Rule is a fundamental rule for differentiating composite functions.
• A composite function is a function composed of two or more functions.
• In simpler terms, if we have a function operating inside another function, we have a composite function.
• The chain rule assists in finding the derivative of such composite functions.

Formula and Method for the Chain Rule

• If we have a composite function y = f(g(x)), the derivative of y with respect to x is dy/dx = f’(g(x)) * g’(x).
• In the above equation, f’(g(x)) is the derivative of the outer function, and g’(x) is the derivative of the inner function.
• This process is known as differentiating from the outside in.

Example of the Chain Rule

• As an illustrative example, let’s take a composite function y = sin(2x).
• The outer function is sin(x) and the inner function is 2x.
• By applying chain rule, the derivative dy/dx of y = sin(2x) is cos(2x) * 2.

Importance of the Chain Rule

• The Chain Rule is essential for differentiating a broad variety of functions.
• It’s part of the foundational toolkit in calculus, and is often used in physical sciences, economics, and engineering.
• It is particularly useful when dealing with functions that are represented as the composition of other functions.

Review of Relevant Formulas

• First derivative using Chain Rule: If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).
• Example with Chain Rule: The derivative of y = sin(2x) is dy/dx = cos(2x) * 2.