Derivatives of tan(x) and cot(x) | Derivative rules | AP Calculus AB | Khan Academy

TL;DR

This video explains how to find the derivatives of tangent and cotangent functions, resulting in secant squared and negative cosecant squared functions, respectively.

Transcript

• [Voiceover] We already know the derivatives of sine and cosine. We know that the derivative with respect to x of sine of x is equal to cosine of x. We know that the derivative with respect to x of cosine of x is equal to negative sine of x. And so what we want to do in this video is find the derivatives of the other basic trig functions. So, in p... Read More

Key Insights

• 😑 The derivative of a function can be found using the quotient rule when it can be expressed as the quotient of two functions.
• 🖐️ The Pythagorean identity plays a crucial role in simplifying trigonometric derivatives.
• ❎ The derivative of the tangent function is secant squared, while the derivative of the cotangent function is negative cosecant squared.
• ❓ Understanding the derivatives of basic trigonometric functions is fundamental in calculus.
• 😒 The use of color coding in the video enhances readability and understanding.
• ⚾ The Pythagorean identity is based on the relationship between the coordinates on the unit circle.
• 😑 The quotient rule is a useful tool for finding derivatives of functions expressed as the quotient of two functions.

Questions & Answers

Q: How do you find the derivative of tangent of x?

To find the derivative of tangent of x, we use the quotient rule. The derivative is equal to secant squared of x.

Q: What is the derivative of cotangent of x?

The derivative of cotangent of x is obtained by applying the quotient rule. The result is negative cosecant squared of x.

Q: What is the Pythagorean identity?

The Pythagorean identity states that the sum of the squares of cosine and sine functions is always equal to one, regardless of the value of x.

Q: How do you simplify the derivative of tangent of x to secant squared of x?

By using the Pythagorean identity, cosine squared of x plus sine squared of x equals one, we can simplify the derivative of tangent of x, resulting in secant squared of x.

Summary & Key Takeaways

• The video teaches how to find the derivative of tangent of x using the quotient rule, resulting in secant squared of x.

• The derivative of cotangent of x is found using the quotient rule, resulting in negative cosecant squared of x.

• The Pythagorean identity, cosine squared of x plus sine squared of x equals one, is derived and used to simplify the results of the derivative calculations.