Derivatives of Trigonometric Functions: 3.5

3.5: Derivatives of Trigonometric Functions

Introduction

This section covers the differentiation rules for the six basic trigonometric functions. Understanding these derivatives is essential for solving problems involving rates of change and for applying calculus to trigonometric models in science and engineering.

Derivatives of the Trigonometric Functions

Basic Derivative Formulas

• Sine: $\frac{d}{dx}(\sin x) = \cos x$

• Cosine: $\frac{d}{dx}(\cos x) = -\sin x$

• Tangent: $\frac{d}{dx}(\tan x) = \sec^2 x$

• Cotangent: $\frac{d}{dx}(\cot x) = -\csc^2 x$

• Secant: $\frac{d}{dx}(\sec x) = \sec x \tan x$

• Cosecant: $\frac{d}{dx}(\csc x) = -\csc x \cot x$

Table: Derivatives of Trigonometric Functions

Function	Derivative
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$

Examples: Differentiating Trigonometric Functions

Example 1: Linear Combinations

Find the derivative of $y = 3\cos x + 5\sin x$.

• Apply the linearity of differentiation:

• $y' = 3(\cos x)' + 5(\sin x)'$

• $= 3(-\sin x) + 5(\cos x)$

• $= -3\sin x + 5\cos x$

Example 2: Difference Involving Cosecant

Find the derivative of $y = 3x - \csc x$.

• $y' = (3x)' - (\csc x)'$

• $= 3 - ( -\csc x \cot x )$

• $= 3 + \csc x \cot x$

Product Rule with Trigonometric Functions

Product Rule Formula

• If $y = f(x)g(x)$, then $y' = f'(x)g(x) + f(x)g'(x)$

Example 3: Exponential and Sine

Find the derivative of $y = e^x \sin x$.

• $y' = (e^x)' \sin x + e^x (\sin x)'$

• $= e^x \sin x + e^x \cos x$

Example 4: Product of Sine and Cosine

Find the derivative of $y = \sin x \cos x$.

• $y' = (\sin x)' \cos x + \sin x (\cos x)'$

• $= \cos x \cos x + \sin x ( -\sin x )$

• $= \cos^2 x - \sin^2 x$

Quotient Rule with Trigonometric Functions

Quotient Rule Formula

• If $y = \frac{f(x)}{g(x)}$, then $y' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Example 5: Quotient with Cosine

Find the derivative of $y = \frac{5\cos x}{7 - 6\cos x}$.

• Apply the quotient rule:

• $y' = \frac{(5\cos x)'(7 - 6\cos x) - 5\cos x (7 - 6\cos x)'}{(7 - 6\cos x)^2}$

• $= \frac{-5\sin x (7 - 6\cos x) - 5\cos x ( -6\sin x )}{(7 - 6\cos x)^2}$

• $= \frac{-35\sin x + 30\sin x \cos x}{(7 - 6\cos x)^2}$

• $= \frac{-35\sin x}{(7 - 6\cos x)^2}$

Example 6: Quotient with Sine

Find the derivative of $y = \frac{x\cos x}{3 + \sin x}$.

• $y' = \frac{(x\cos x)'(3 + \sin x) - x\cos x (3 + \sin x)'}{(3 + \sin x)^2}$

• $(x\cos x)' = \cos x - x\sin x$ (by product rule)

• $(3 + \sin x)' = \cos x$

• Substitute and simplify:

• $y' = \frac{(\cos x - x\sin x)(3 + \sin x) - x\cos x \cos x}{(3 + \sin x)^2}$

• $= \frac{3\cos x + \cos x \sin x - 3x\sin x - x\sin^2 x - x\cos^2 x}{(3 + \sin x)^2}$

Higher-Order Derivatives

Example 7: Second Derivative of a Secant Function

Find the second derivative of $y = 5\sec x$.

• First derivative: $y' = 5(\sec x)' = 5\sec x \tan x$

• Second derivative: $y'' = 5(\sec x \tan x)'$

• Apply the product rule:

• $= 5[ (\sec x)' \tan x + \sec x (\tan x)' ]$

• $= 5[ \sec x \tan x \tan x + \sec x \sec^2 x ]$

• $= 5\sec x \tan^2 x + 5\sec^3 x$

Summary Table: Product and Quotient Rules

Rule	Formula
Product Rule	$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
Quotient Rule	$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Key Points

• Memorize the derivatives of all six trigonometric functions.

• Apply the product and quotient rules when differentiating combinations of trigonometric functions with other functions.

• For higher-order derivatives, repeatedly apply the differentiation rules as needed.