Product rule proof | Taking derivatives | Differential Calculus | Khan Academy

TL;DR

This video provides a step-by-step proof of the product rule for finding derivatives.

Transcript

• [Voiceover] What I hope to do in this video is give you a satisfying proof of the product rule. So let's just start with our definition of a derivative. So if I have the function F of X, and if I wanted to take the derivative of it, by definition, by definition, the derivative of F of X is the limit as H approaches zero, of F of X plus H minus F ... Read More

Key Insights

• ⛔ The derivative is defined as the limit of the difference quotient.
• 👻 The product rule allows for finding the derivative of the product of two functions.
• 😑 Manipulating expressions through addition and subtraction can lead to algebraic simplifications.
• 📏 The product rule provides a formula for finding the derivative of a product of functions.
• 📏 The proof of the product rule involves using the limit properties and the definition of the derivative.
• 📏 The product rule is a fundamental concept in calculus.
• ⌛ The derivative of a function times another function is equal to one function times the derivative of the other plus the other function times the derivative of the first.

Questions & Answers

Q: What is the definition of the derivative?

The derivative of a function F(X) is the limit as H approaches zero of [F(X + H) - F(X)] / H.

Q: How is the product rule derived from the definition of the derivative?

By applying the definition of the derivative to F(X) times G(X) and manipulating the expression algebraically, the product rule is obtained.

Q: What are the components of the product rule?

The product rule states that the derivative of F(X) times G(X) equals F(X) times the derivative of G(X) plus G(X) times the derivative of F(X).

Q: Is the product rule applicable to all functions?

Yes, the product rule can be used to find the derivative of any product of two functions.

Summary & Key Takeaways

• The video presents a definition of the derivative and explains how to find the derivative of a function.

• The video then focuses on finding the derivative of the product of two functions and introduces the concept of the product rule.

• By manipulations and additions/subtractions, the video shows how the product rule can be derived from the definition of the derivative.