How to Find Derivatives of Algebraic Functions Easily

Master trigonometric derivatives like sin, cos, and tan with clear rules, solved examples, and practice problems for high school calculus learners.

📘 Derivatives of Trigonometric Functions

Master Derivatives of Sine, Cosine, and More — With Easy Rules and Examples

🧭 Introduction: Trigonometry Meets Calculus

Trigonometric functions describe waves, circular motion, and periodic behavior — from sound vibrations to tides. Understanding their derivatives helps us model real-world change.

This article will walk you through:

• Basic derivatives of trig functions
• Rules for combined and composite functions
• Applications with practice and HOTS-level problems

🎯 Basic Derivatives of Trigonometric Functions

Let’s start with the standard six trig functions:

These are essential. Memorize them early — they’re used often in exams and applications.

🧩 Applying the Chain Rule

When a trig function has something more complex inside it (like $\sin(3x)$ or $\cos(x^2)$), we use the chain rule:

If $f(x) = \sin(g(x))$, then:

$$ f'(x) = \cos(g(x)) \cdot g'(x) $$

🔁 Example 1: Derivative of $\sin(3x)$

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

• Outer: $\sin(u) \Rightarrow \cos(u)$
• Inner: $u = 3x \Rightarrow u' = 3$

Then:

$$ f'(x) = \cos(3x) \cdot 3 = 3\cos(3x) $$

🔁 Example 2: Derivative of $\cos(x^2)$

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

• Outer: $\cos(u) \Rightarrow -\sin(u)$
• Inner: $u = x^2 \Rightarrow u' = 2x$

Then:

$$ f'(x) = -\sin(x^2) \cdot 2x = -2x\sin(x^2) $$

🔁 Using Product and Quotient Rules with Trig

✴️ Example 3: Product Rule

Let $f(x) = x \cdot \cos x$

Then:

$$ f'(x) = (1)(\cos x) + x(-\sin x) = \cos x - x\sin x $$

✴️ Example 4: Quotient Rule

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

Then:

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

This appears in physics and engineering when analyzing oscillations and waves.

🌊 Application in Real Life: Sound Waves

The position of a vibrating string is modeled by:

$$ s(t) = 5\sin(2\pi t) $$

To find the velocity, take the derivative:

$$ s'(t) = 5 \cdot 2\pi \cos(2\pi t) = 10\pi \cos(2\pi t) $$

This gives the instantaneous velocity of the vibration.

🧠 HOTS Problem: Trig & Chain Rule

Let $f(x) = \tan(x^2 + 1)$

Find $f'(x)$

🧠 Solution Sketch:

• Outer: $\tan(u) \Rightarrow \sec^2(u)$
• Inner: $u = x^2 + 1 \Rightarrow u' = 2x$

So:

$$ f'(x) = \sec^2(x^2 + 1) \cdot 2x = 2x \sec^2(x^2 + 1) $$

🧪 Practice Problems (Standard)

Differentiate the following:

1. $f(x) = \cos x$
2. $f(x) = \tan(3x)$
3. $f(x) = \frac{\sin x}{x}$
4. $f(x) = x^2 \cdot \cos x$
5. $f(x) = \cot(x^3)$

🔥 HOTS Practice Problems

🚀 Problem 1:

A pendulum swings with angle modeled by:

$$ \theta(t) = 4\cos(\sqrt{t}) $$

Find the angular velocity $\theta'(t)$

🚀 Problem 2:

Let $f(x) = \frac{\tan x}{x^2 + 1}$

Find $f'(x)$ and identify where it is zero on the interval $[0, \pi/2)$

🎯 Summary Table