1. What is the Chain Rule?

The chain rule is a formula for computing the derivative of the composition of two or more functions. It's one of the most important rules in differential calculus.

2. How Does the Chain Rule Work?

The chain rule formula is:

Where:

• \( f(x) \) — The outer function
• \( g(x) \) — The inner function
• \( f'(x) \) — Derivative of the outer function
• \( g'(x) \) — Derivative of the inner function

Explanation: The chain rule tells us that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

3. Importance of the Chain Rule

Details: The chain rule is essential for differentiating complex functions that are composed of simpler functions. It's widely used in physics, engineering, economics, and other fields where rates of change need to be calculated.

4. Using the Calculator

Tips: Enter the outer function f(x) and inner function g(x) in terms of x. The calculator will show how to apply the chain rule to find the derivative.

5. Frequently Asked Questions (FAQ)

Q1: When should I use the chain rule?
A: Use the chain rule whenever you need to differentiate a function that is composed of other functions (a function of a function).

Q2: Can the chain rule be extended to more than two functions?
A: Yes, for three functions it would be: d/dx[f(g(h(x)))] = f'(g(h(x)))·g'(h(x))·h'(x)

Q3: What are some common examples of chain rule applications?
A: Exponential functions (e^u), trigonometric functions (sin(u)), logarithmic functions (ln(u)), etc. where u is a function of x.

Q4: How is the chain rule related to u-substitution in integration?
A: U-substitution is essentially the chain rule in reverse, used for integrating composite functions.

Q5: What's the difference between chain rule and product rule?
A: Chain rule is for composite functions (functions of functions), while product rule is for products of functions.