Derivatives of Trigonometric Functions

By the end of this lesson, you will be able to differentiate all six trig functions, prove the derivative of sine and cosine from the limit definition, and apply these in combination with the product, quotient, and chain rules.

The Six Trig Derivatives

$$\frac{d}{dx}[\sin x] = \cos x \qquad \frac{d}{dx}[\cos x] = -\sin x$$

$$\frac{d}{dx}[\tan x] = \sec^2 x \qquad \frac{d}{dx}[\cot x] = -\csc^2 x$$

$$\frac{d}{dx}[\sec x] = \sec x\tan x \qquad \frac{d}{dx}[\csc x] = -\csc x\cot x$$

Memory pattern: derivatives of co-functions (cos, cot, csc) all carry a negative sign.

Proof: $(\sin x)' = \cos x$

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

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

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

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

Proof: $(\tan x)' = \sec^2 x$

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

Example 1 — Direct Application

$f(x) = 3\sin x - 5\cos x + \tan x$

$f'(x) = 3\cos x + 5\sin x + \sec^2 x$

Example 2 — Product with Trig

$g(x) = x^2\sin x$

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

Example 3 — Quotient with Trig

$h(x) = \dfrac{\sin x}{1 + \cos x}$

$h'(x) = \dfrac{\cos x(1+\cos x) - \sin x(-\sin x)}{(1+\cos x)^2}$

$= \dfrac{\cos x + \cos^2 x + \sin^2 x}{(1+\cos x)^2} = \dfrac{\cos x + 1}{(1+\cos x)^2} = \dfrac{1}{1+\cos x}$

Example 4 — Second Derivative of Sine

$$\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d^2}{dx^2}[\sin x] = -\sin x$$

$\sin x$ satisfies the differential equation $y'' = -y$. This is the foundation of oscillatory motion in physics.

Trig Derivatives at Specific Points

$\dfrac{d}{dx}[\sin x]\bigg|_{x=\pi/6} = \cos(\pi/6) = \dfrac{\sqrt{3}}{2}$

$\dfrac{d}{dx}[\tan x]\bigg|_{x=\pi/4} = \sec^2(\pi/4) = 2$

Summary