Derivatives of All Six Trigonometric Functions

Trigonometric Derivatives

By the end of this lesson, you will be able to differentiate all six trigonometric functions from memory and apply them in combination with other rules.

Derivatives of the Six Trig Functions

These must be memorized:

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

Memory patterns:

• The "co-" functions (cosine, cosecant, cotangent) all have a negative sign in their derivative
• $\tan x$ → $\sec^2 x$ and $\cot x$ → $-\csc^2 x$ are "squared" forms
• $\sec x$ and $\csc x$ keep themselves as a factor in their own derivative

Deriving sin x and cos x from the Limit Definition

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

Using the sum identity $\sin(x+h) = \sin x\cos h + \cos x\sin h$:

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

$$= \sin x \cdot \underbrace{\lim_{h\to 0}\frac{\cos h - 1}{h}}{= 0} + \cos x \cdot \underbrace{\lim{h\to 0}\frac{\sin h}{h}}_{= 1} = \cos x$$

The key special limits: $\lim_{h\to 0}\frac{\sin h}{h} = 1$ and $\lim_{h\to 0}\frac{\cos h - 1}{h} = 0$.

Deriving the Other Trig Derivatives

$\tan x = \frac{\sin x}{\cos x}$ → apply the quotient rule:

$$\frac{d}{dx}[\tan x] = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$$

Examples

Example 1: Differentiate $f(x) = 3\sin x - 2\cos x + \tan x$

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

Example 2: Differentiate $g(x) = x^2\sin x$

Product rule: $g'(x) = 2x\sin x + x^2\cos x$

Example 3: Differentiate $h(x) = \dfrac{\tan x}{x}$

Quotient rule: $h'(x) = \dfrac{\sec^2 x \cdot x - \tan x \cdot 1}{x^2} = \dfrac{x\sec^2 x - \tan x}{x^2}$