D - Derivatives of Basic Trigonometric Functions Lesson

Derivatives of Basic Trigonometric Functions

As we begin the study of derivatives of trigonometric functions, recall that values of each of the trigonometric functions sin x, cos x, tan x, cot x, sec x and csc x are based on an angle whose radian measure is x. Using degree measures does not yield the expected answers. Please note we will use radian measures exclusively for the remainder of the course. Your graphing calculator should also be set in radian mode.

Derivatives of Sine and Cosine Functions

An intuitive approach to determining the derivative of y = sin x and y = cos x using graphical and numerical methods and the TI-84 is illustrated below. Of course, such approaches are not legitimate proofs as they only suggest likely possibilities.

Derivatives of Other Basic Trigonometric Functions

Derivative formulas for tan x, cot x, sec x, and csc x are developed from using related trigonometric functions involving sin x and cos x and applying the Quotient Rule.

In summary, derivatives of the six trigonometric functions are given below.

equation image indicator $LaTeX: \frac{d}{dx}\left(\sin x\right)=\cos x\\ \frac{d}{dx}\left(\cos x\right)=-\sin x\\ \frac{d}{dx}\left(\tan x\right)=\sec^2 x\\ \frac{d}{dx}\left(\cot x\right)=-\csc^2 x\\ \frac{d}{dx}\left(\sec x\right)=(\sec x)(\tan x)\\ \frac{d}{dx}\left(\csc x\right)=-(\csc x)(\cot x)$ $\frac{d}{dx}\left(\sin x\right)=\cos x\\ \frac{d}{dx}\left(\cos x\right)=-\sin x\\ \frac{d}{dx}\left(\tan x\right)=\sec^2 x\\ \frac{d}{dx}\left(\cot x\right)=-\csc^2 x\\ \frac{d}{dx}\left(\sec x\right)=(\sec x)(\tan x)\\ \frac{d}{dx}\left(\csc x\right)=-(\csc x)(\cot x)$

Notice the negative signs in the derivative formulas for the cofunctions.

Derivatives of Basic Trigonometric Functions Practice

Find the derivative.

Derivatives of Basic Trigonometric Functions: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of derivatives of basic trigonometric functions.

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