1. What is the Chain Rule?

The chain rule is a fundamental theorem in calculus for differentiating composite functions. It allows us to find the derivative of a function that is composed of other functions.

2. How Does the Chain Rule Work?

The chain rule formula is:

Where:

• \( \frac{dy}{dx} \) — Derivative of y with respect to x
• \( \frac{dy}{du} \) — Derivative of y with respect to u
• \( \frac{du}{dx} \) — Derivative of u with respect to x

Explanation: The chain rule breaks down the differentiation of complex functions into simpler parts by introducing an intermediate variable (u).

3. Importance of the Chain Rule

Details: The chain rule is essential for differentiating composite functions that appear frequently in physics, engineering, economics, and other sciences. It's one of the most important rules in differential calculus.

4. Using the Calculator

Tips: Enter the derivatives dy/du and du/dx. The calculator will compute dy/dx by multiplying these two values together.

5. Frequently Asked Questions (FAQ)

Q1: When should I use the chain rule?
A: Use the chain rule whenever you need to differentiate a composite function (a function of a function).

Q2: What's an example of a chain rule application?
A: If y = sin(x²), then dy/dx = cos(x²) × 2x. Here, u = x², dy/du = cos(u), and du/dx = 2x.

Q3: Can the chain rule be extended to more functions?
A: Yes, for y = f(g(h(x))), the chain rule becomes dy/dx = f'(g(h(x))) × g'(h(x)) × h'(x).

Q4: How is the chain rule related to the product rule?
A: While both are differentiation rules, the product rule is for products of functions, while the chain rule is for compositions of functions.

Q5: Is there a version for partial derivatives?
A: Yes, the multivariable chain rule extends this concept to functions of several variables.