Derivative of inverse cosine | Taking derivatives | Differential Calculus | Khan Academy

TL;DR

The derivative of the inverse cosine of x is equal to -1 over the square root of 1 minus x squared.

Transcript

Voiceover: In the last video, we showed or we proved to ourselves that the derivative of the inverse sine of x is equal to 1 over the square root of 1 minus x squared. What I encourage you to do in this video is to pause it and try to do the same type of proof for the derivative of the inverse cosine of x. So, our goal here is to figure out ... I w... Read More

Key Insights

• 🙃 The derivative of the inverse cosine of x is derived by differentiating both sides of the equation.
• 😑 Rewriting the expression in terms of cosine allows for a simplified form of the derivative.
• 👨‍💼 The derivative of the inverse cosine is negative, while the derivative of the inverse sine is positive.

Questions & Answers

Q: How is the derivative of the inverse cosine of x derived?

By setting y as the inverse cosine of x and differentiating both sides of the equation, we find that the derivative is -1 over the square root of 1 minus x squared.

Q: Why is it important to rewrite the expression in terms of cosine instead of sine?

It is important to rewrite the expression in terms of cosine instead of sine because x is equal to the cosine of y. This allows us to simplify the expression and obtain a derivative in terms of x.

Q: What is the difference between the derivative of the inverse cosine and the derivative of the inverse sine?

The only difference between the two derivatives is the sign. The derivative of the inverse cosine is negative, while the derivative of the inverse sine is positive.

Q: What is the significance of knowing the derivative of the inverse cosine?

Knowing the derivative of the inverse cosine is useful in calculus and other mathematical applications where finding the rate of change of inverse cosine functions is necessary.

Summary & Key Takeaways

• The video demonstrates a proof for finding the derivative of the inverse cosine of x.

• By setting y as the inverse cosine of x, the derivative is found by differentiating both sides of the equation.

• The final expression for the derivative is -1 over the square root of 1 minus x squared.