Quotient rule and common derivatives | Taking derivatives | Differential Calculus | Khan Academy

TL;DR

This video explains how to find the derivative of complex expressions using the product rule and introduces common derivatives such as e to the x and natural log of x.

Transcript

Welcome back. Let's do some more derivative problems. Let's say I want to figure out the derivative d over dx of-- and let me give something that looks a little bit different-- x to the third minus 5x to the fifth, all of that to the third power over 2x plus 5 to the fifth power. This is a parentheses. This is just saying that I want to take the de... Read More

Key Insights

• 😑 The product rule is a useful tool for finding the derivative of complex expressions.
• ☺️ The derivative of e to the x is e to the x itself, which is a unique and interesting property of e.
• ☺️ The derivative of the natural log of x is 1/x, providing a surprising connection between logarithms and derivatives.
• ❓ Common derivatives of trigonometric functions are sin(x), cos(x), and tan(x), which are important to memorize for calculus problems.
• 📈 It is recommended to graph functions and their derivatives to understand the relationship between slopes and the original function.
• ❓ Memorizing derivatives can greatly facilitate solving derivative problems efficiently.

Questions & Answers

Q: How do you find the derivative of a complex expression using the product rule?

The product rule allows us to find the derivative of a product of two functions. We differentiate one function and multiply it by the original form of the second function, then differentiate the second function and multiply it by the original form of the first function. Finally, we add the two results together.

Q: What is the derivative of e to the x?

The derivative of e to the x is e to the x itself. This means that at any point along the curve of e to the x, the slope of that point is equal to e to the x.

Q: How do you find the derivative of the natural log of x?

The derivative of the natural log of x is 1/x. This is a remarkable property as it shows that the slope of the natural log function at any point is equal to 1 divided by x.

Q: What are the common derivatives of trigonometric functions?

The derivative of sin(x) is cos(x), the derivative of cos(x) is -sin(x), and the derivative of tan(x) is 1/cos^2(x) or sec^2(x).

Summary & Key Takeaways

• The video begins with a demonstration of finding the derivative of a complex expression using the product rule.

• It then introduces and proves two key derivatives: the derivative of e to the x is e to the x, and the derivative of the natural log of x is 1/x.

• Additional common derivatives are mentioned, such as the derivative of sin(x) is cos(x), cosine of x is -sin(x), and tan(x) is 1/cos^2(x).