Derivatives of Trigonometric Functions

Trigonometric functions are fundamental in calculus. Their derivatives follow simple patterns that repeat and are best remembered as a cycle. The chain rule allows you to handle more complex arguments.

📜 Core Formulas

$$ \frac{d}{dx}\big(\sin x\big) = \cos x $$ $$ \frac{d}{dx}\big(\cos x\big) = -\sin x $$ $$ \frac{d}{dx}\big(\tan x\big) = \frac{1}{\cos^2 x} = \sec^2 x \quad (x \neq \tfrac{\pi}{2}+k\pi) $$

💡 Cycle: Differentiating sine four times brings you back to sine. 💡 Domain: For $\tan(x)$, remember the function is undefined at $x = \tfrac{\pi}{2}+k\pi$.

🪜 Application Strategy

Identify the trig function: Is it $\sin$, $\cos$, $\tan$, or a combination?
Apply the derivative rule: Use the core formulas above.
Use the chain rule if needed: If the argument is $u(x)$, then $$ \frac{d}{dx}(\sin u) = \cos(u)\,u'(x), \quad \frac{d}{dx}(\cos u) = -\sin(u)\,u'(x), \quad \frac{d}{dx}(\tan u) = \frac{u'(x)}{\cos^2(u)}. $$

✏️ Examples

Example 1: Simple Sine

$f(x) = \sin x$

Derivative: $$ f'(x) = \cos x. $$

Example 2: Simple Cosine

$f(x) = \cos x$

Derivative: $$ f'(x) = -\sin x. $$

Example 3: Simple Tangent

$f(x) = \tan x$

Derivative: $$ f'(x) = \frac{1}{\cos^2 x}. $$

Example 4: Chain Rule with Sine

$f(x) = \sin(3x)$

Outer derivative: $\cos(3x)$
Inner derivative: $3$

Result: $$ f'(x) = 3\cos(3x). $$

Example 5: Chain Rule with Cosine

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

Outer derivative: $-\sin(x^2)$
Inner derivative: $2x$

Result: $$ f'(x) = -2x\sin(x^2). $$

Example 6: Chain Rule with Tangent

$f(x) = \tan(5x-1)$

Outer derivative: $\sec^2(5x-1) = \dfrac{1}{\cos^2(5x-1)}$
Inner derivative: $5$

Result: $$ f'(x) = \frac{5}{\cos^2(5x-1)}. $$