#Derivatives of Sine and Cosine Functions

#Table of Contents

1. Introduction
2. Differentiating Sin x and Cos x
3. Differentiating Sin kx and Cos kx
4. Worked Examples
5. Using the Definition of a Derivative
6. Glossary
7. Practice Questions
8. Summary and Key Takeaways

#Introduction

In calculus, understanding how to differentiate sine and cosine functions is fundamental. This guide covers the rules for differentiating these trigonometric functions, including their more complex forms, and provides worked examples and practice questions to help solidify your understanding.

#Differentiating Sin x and Cos x

#Key Concepts

• If $f(x) = \sin x$,

• If $g(x) = \cos x$,

• If $h(x) = -\sin x$,

• If $j(x) = -\cos x$,

• This sequence then repeats.

#Differentiating Sin kx and Cos kx

When dealing with functions of the form $\sin(kx)$ and $\cos(kx)$, the differentiation involves the chain rule.

#Key Concepts

• If $f(x) = \sin(kx)$,

• If $g(x) = \cos(kx)$,

#Worked Examples

#Example 1

Differentiate the following function: $$f(x) = \cos x - 4 \sin x$$

Solution: $$\cos x$$ differentiates to $$-\sin x$$
$$\sin x$$ differentiates to $$\cos x$$

So, $$f'(x) = -\sin x - 4 \cos x$$

#Example 2

Differentiate the following function: $$g(x) = 9 \sin \left(\frac{2}{3} x\right) - 2 \cos (3x)$$

Solution: $$\sin(kx)$$ differentiates to $$k \cos(kx)$$
$$\cos(kx)$$ differentiates to $$-k \sin(kx)$$

So, $$g'(x) = 9 \cdot \frac{2}{3} \cos \left(\frac{2}{3} x\right) - 2 \left(-3 \sin(3x)\right)$$

Simplifying, $$g'(x) = 6 \cos \left(\frac{2}{3} x\right) + 6 \sin(3x) = 6\left(\cos \left(\frac{2}{3} x\right) + \sin(3x)\right)$$

#Using the Definition of a Derivative

#Definition

The derivative of a function $f(x)$ can be defined as: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

#Differentiating $\sin x$ Using the Definition

Given $f(x) = \sin x$: $$f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}$$

Using the addition formula: $$\sin(A+B) = \sin A \cos B + \cos A \sin B$$

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

Factorizing: $$f'(x) = \lim_{h \to 0} \left(\sin x \left(\frac{\cos h - 1}{h}\right) + \cos x \left(\frac{\sin h}{h}\right)\right)$$

Applying the limits: $$\lim_{h \to 0} \frac{\sin h}{h} = 1$$ and $$\lim_{h \to 0} \frac{\cos h - 1}{h} = 0$$

So, $$f'(x) = \sin x \cdot 0 + \cos x \cdot 1 = \cos x$$

#Differentiating $\cos x$ Using the Definition

Given $g(x) = \cos x$: $$g'(x) = \lim_{h \to 0} \frac{\cos(x+h) - \cos(x)}{h}$$

Using the addition formula: $$\cos(A+B) = \cos A \cos B - \sin A \sin B$$

So, $$g'(x) = \lim_{h \to 0} \frac{\cos x \cos h - \sin x \sin h - \cos x}{h}$$

Factorizing: $$g'(x) = \lim_{h \to 0} \left(\cos x \left(\frac{\cos h - 1}{h}\right) - \sin x \left(\frac{\sin h}{h}\right)\right)$$

Applying the limits: $$\lim_{h \to 0} \frac{\sin h}{h} = 1$$ and $$\lim_{h \to 0} \frac{\cos h - 1}{h} = 0$$

So, $$g'(x) = \cos x \cdot 0 - \sin x \cdot 1 = -\sin x$$

#Glossary

• Derivative: A measure of how a function changes as its input changes.
• Chain Rule: A formula to compute the derivative of a composite function.
• Trigonometric Addition Formulae: Formulas that express trigonometric functions of sums of angles in terms of products of trigonometric functions of individual angles.
• Limit: The value that a function approaches as the input approaches some value.

#Practice Questions

Practice Question

1. Differentiate $f(x) = 5 \sin(2x)$.

2. Differentiate $h(x) = 4 \cos(3x) - \sin(5x)$.

3. Use the definition of a derivative to differentiate $g(x) = \sin x$.

#Summary and Key Takeaways

• The derivative of $\sin x$ is $\cos x$.
• The derivative of $\cos x$ is $-\sin x$.
• Using the chain rule, the derivatives of $\sin(kx)$ and $\cos(kx)$ are $k \cos(kx)$ and $-k \sin(kx)$, respectively.
• Understanding and applying the definition of a derivative is crucial for deriving these results.

#Key Takeaways

For more complex functions, break them down into simpler components and apply differentiation rules step by step.