1. What is the Chain Rule?

The chain rule is a fundamental theorem in calculus for differentiating composite functions. It states that the derivative of a composite function is the product of the derivatives of the constituent functions.

2. How Does the Calculator Work?

The calculator uses the chain rule formula:

Where:

• \( \frac{dy}{du} \) — Derivative of the outer function
• \( \frac{du}{dx} \) — Derivative of the inner function

Explanation: The chain rule allows us to break down complex derivatives into simpler components that can be multiplied together.

3. Importance of the Chain Rule

Details: The chain rule is essential for differentiating a wide variety of functions in physics, engineering, economics, and other fields where composite functions appear.

4. Using the Calculator

Tips: Enter the derivatives of the outer and inner functions. The calculator will multiply them together to give the final 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 composite function (a function of another function).

Q2: What's an example of chain rule application?
A: For sin(x²), dy/du = cos(u) where u = x², and du/dx = 2x. The derivative is cos(x²)*2x.

Q3: Can the chain rule be extended to more functions?
A: Yes, for f(g(h(x))), the derivative would be f'(g(h(x))) * g'(h(x)) * h'(x).

Q4: How does this relate to the product rule?
A: The product rule is for products of functions, while the chain rule is for compositions of functions.

Q5: What if my function is very complex?
A: Break it down into simpler components and apply the chain rule multiple times if needed.