What Are the Derivatives of Tan(x) and Cot(x)?

TL;DR

The derivatives of the tangent and cotangent functions are secant squared and negative cosecant squared, respectively. This video demonstrates the application of the quotient rule and the Pythagorean identity to derive these results systematically.

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.

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.