Derivatives of Trigonometric Functions

Basic Trigonometric Derivatives

Memory Aids

• Derivatives of "co" functions ($\cos$, $\cot$, $\csc$) have negative signs
• $\sin$ and $\cos$ cycle: $\sin \to \cos \to -\sin \to -\cos \to \sin$
• $\tan$ and $\sec$ are related: $(\tan)' = \sec^2$, $(\sec)' = \sec \cdot \tan$
• $\cot$ and $\csc$ are related: $(\cot)' = -\csc^2$, $(\csc)' = -\csc \cdot \cot$

With Chain Rule

For $u = u(x)$:

Proofs Using Limit Definition

Derivative of sin(x)

$\frac{d}{dx}[\sin(x)] = \lim_{h \to 0} \frac{\sin(x + h) - \sin(x)}{h}$

Using $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$:

$= \lim_{h \to 0} \frac{\sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)}{h}$

$= \lim_{h \to 0} \sin(x)\frac{\cos(h) - 1}{h} + \cos(x)\frac{\sin(h)}{h}$

$= \sin(x) \cdot 0 + \cos(x) \cdot 1 = \cos(x)$

Examples

Example 1: Basic derivatives

$f(x) = 3\sin(x) - 2\cos(x)$

$f'(x) = 3\cos(x) + 2\sin(x)$

Example 2: With chain rule

$f(x) = \sin(3x^2 + 1)$

$f'(x) = \cos(3x^2 + 1) \cdot 6x = 6x \cos(3x^2 + 1)$

Example 3: Product rule with trig

$f(x) = x^2 \tan(x)$

$f'(x) = x^2 \cdot \sec^2(x) + \tan(x) \cdot 2x = x^2 \sec^2(x) + 2x \tan(x)$

Example 4: Quotient with trig

$f(x) = \frac{\sin(x)}{x}$

$f'(x) = \frac{x \cdot \cos(x) - \sin(x) \cdot 1}{x^2} = \frac{x \cos(x) - \sin(x)}{x^2}$

Example 5: Nested trig functions

$f(x) = \cos(\sin(x))$

$f'(x) = -\sin(\sin(x)) \cdot \cos(x)$

Example 6: sec and tan

$f(x) = \sec(2x) \tan(2x)$

$f'(x) = \sec(2x) \cdot \sec^2(2x) \cdot 2 + \tan(2x) \cdot \sec(2x)\tan(2x) \cdot 2$

$= 2\sec^3(2x) + 2\sec(2x)\tan^2(2x)$

$= 2\sec(2x)[\sec^2(2x) + \tan^2(2x)]$