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

TL;DR

The video explains how to find the derivative of the inverse tangent function using trigonometric identities.

Transcript

We already know that the derivative with respect to x of tangent of x is equal to the secant of x squared, which is of course the same thing of one over cosine of x squared. Now what we wanna do in this video, like we've done in the last few videos, is figure out what the derivative of the inverse function of the tangent of x is, or in particular, ... Read More

Key Insights

• 🌆 The derivative of the inverse tangent function can be found by setting y equal to inverse tangent of x.
• 🟰 The chain rule is applied to find the derivative of tangent of y, which is equal to x.
• ❣️ The derivative of y with respect to x is expressed as cosine of y squared.

Questions & Answers

Q: What is the derivative of the inverse tangent function?

The derivative of the inverse tangent function is one over one plus x squared.

Q: How is the derivative derived using trigonometric identities?

The video uses the Pythagorean identity and the relationship between sine, cosine, and tangent to derive the derivative expression.

Q: Why is it important to express the derivative as a function of x?

Expressing the derivative as a function of x allows for easy calculation and understanding of the rate of change of the inverse tangent function.

Q: What is the chain rule and how is it applied in this context?

The chain rule allows for the calculation of the derivative of a composite function. In this case, it is used to find the derivative of the tangent of y, which is a function of x.

Summary & Key Takeaways

• The video discusses finding the derivative of the inverse tangent function by setting y equal to inverse tangent of x.

• After applying the chain rule and simplifying, the derivative of y with respect to x is found to be cosine of y squared.

• By expressing cosine of y squared as a function of tangent of y, the derivative can be written as one over one plus x squared.