Basic Trig Derivatives Explained: Why is the derivative of sin x cos x?

📚 Why is the Derivative of $sin(x)$ equal to $cos(x)$?

The derivative of $sin(x)$ being $cos(x)$ is a fundamental concept in calculus. Let's break down the 'why' behind it, exploring its definition, historical context, key principles, and practical examples.

📜 A Glimpse into History

The development of calculus, including derivatives, is largely attributed to Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. They provided frameworks for understanding rates of change, which led to the formulation of derivative rules. Understanding trigonometric derivatives was crucial for modelling periodic phenomena.

🗝️ Key Principles: Defining the Derivative

• 🔎 The Derivative: The derivative, denoted as $\frac{d}{dx}f(x)$, measures the instantaneous rate of change of a function $f(x)$. It's formally defined as: $$ \frac{d}{dx}f(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$
• 📐 Angle Sum Identity: We'll use the trigonometric identity: $sin(a + b) = sin(a)cos(b) + cos(a)sin(b)$.
• 🌱 Limit Rules: We need to recall two crucial limits: $\lim_{h \to 0} \frac{sin(h)}{h} = 1$ and $\lim_{h \to 0} \frac{cos(h) - 1}{h} = 0$. These are often proven geometrically or using L'Hôpital's Rule.

📝 The Proof

Let's apply the definition of the derivative to $f(x) = sin(x)$:

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

Using the angle sum identity:

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

Rearrange the terms:

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

Separate the limit:

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

Since $sin(x)$ and $cos(x)$ do not depend on $h$, we can take them out of the limit:

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

Now, apply the known limits:

$$ = sin(x) * 0 + cos(x) * 1 $$

Therefore:

$$ \frac{d}{dx}sin(x) = cos(x) $$

🌍 Real-World Examples

• 🌊 Modeling Waves: Trigonometric functions are used to model wave phenomena (sound waves, light waves, water waves). The derivative, $cos(x)$, describes the rate of change of these waves, critical in fields like physics and engineering.
• ⚙️ Simple Harmonic Motion: The motion of a pendulum or a mass on a spring can be described using $sin(x)$ and $cos(x)$. Their derivatives relate to the velocity and acceleration of the object.
• 📈 Signal Processing: In signal processing, trigonometric functions are used to analyze and synthesize signals. The derivatives are essential for understanding how signals change over time.

💡 Conclusion

The derivative of $sin(x)$ being $cos(x)$ isn't just a mathematical curiosity. It's a fundamental relationship with deep connections to how we model and understand the world around us. By understanding the formal definition of the derivative, leveraging trigonometric identities, and applying limit rules, we can clearly see why this relationship holds true.