## Introduction

Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. The derivative of a function provides the rate at which the function is changing at any given point. This concept is widely used in physics, engineering, economics, biology, and many other fields.

## Basic Definition

For a function \(f(x)\), the derivative, often denoted by \(f'(x)\) or \(\frac{df}{dx}\), gives the slope of the tangent line to the curve represented by the function at any given point \(x\).

### The Difference Quotient:

The derivative is often defined using the difference quotient:

\(f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h}\)

## Basic Differentiation Rules

### 1. Constant Rule:

The derivative of a constant is zero.

\(\frac{d}{dx}c = 0\) (where \(c\) is a constant)

### 2. Power Rule:

For any real number \(n\),

\(\frac{d}{dx}x^n = nx^{n-1}\)

### 3. Constant Multiple Rule:

If \(c\) is a constant and \(f(x)\) is a function, then

\(\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)\)

### 4. Sum/Difference Rule:

For functions \(f(x)\) and \(g(x)\),

\(\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\)

### 5. Product Rule:

For functions \(f(x)\) and \(g(x)\),

\(\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)\)

### 6. Quotient Rule:

For functions \(f(x)\) and \(g(x)\) where \(g(x) \neq 0\),

\(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2}\)

## Some Common Derivatives

- \(\frac{d}{dx} \sin(x) = \cos(x)\)
- \(\frac{d}{dx} \cos(x) = -\sin(x)\)
- \(\frac{d}{dx} \tan(x) = \sec^2(x)\)
- \(\frac{d}{dx} \ln(x) = \frac{1}{x}\)
- \(\frac{d}{dx} e^x = e^x\)

## Chain Rule

The chain rule is used to differentiate composite functions. If we have a function \(u(x)\) inside another function \(f(u)\), then:

\(\frac{d}{dx}[f(u(x))] = f'(u) \cdot u'(x)\)

### Example:

To differentiate \(f(x) = \sin(3x^2 + 2x)\):

Here, \(u = 3x^2 + 2x\) and \(f(u) = \sin(u)\).

Using the chain rule,

\(f'(x) = \cos(3x^2 + 2x) \cdot (6x + 2)\)

### Basic Differentiation of Power Functions:

- Differentiate \(f(x) = x^3 + 4x^2 – 7x\). Using the power rule \(\frac{d}{dx} x^n = nx^{n-1}\):

\(f'(x) = 3x^2 + 8x – 7\)

Differentiate \(g(x) = 2x^4 – 5x^3 + x^2\).

\(g'(x) = 8x^3 – 15x^2 + 2x\)

Differentiate \(h(x) = -x^5 + 3x^3 – 2x\).

\(h'(x) = -5x^4 + 9x^2 – 2\)

Differentiate \(j(x) = x^7 + 4x^6 – x^4 + 3\).

\(j'(x) = 7x^6 + 24x^5 – 4x^3\)

### Basic Differentiation for Other Rules:

**Differentiation of Constants**: Differentiate \(k(x) = 5\). Since the derivative of a constant is zero:

\(k'(x) = 0\)

**Differentiation of Exponential Functions**:

Differentiate \(f(x) = e^x\).

Using the rule \(\frac{d}{dx} e^x = e^x\):

\(f'(x) = e^x\)

**Differentiation of Trigonometric Functions**:

a) Differentiate \(g(x) = \sin(x)\).

Using the rule \(\frac{d}{dx} \sin(x) = \cos(x)\):

\(g'(x) = \cos(x)\)

b) Differentiate \(h(x) = \cos(x)\).

Using the rule \(\frac{d}{dx} \cos(x) = -\sin(x)\):

\(h'(x) = -\sin(x)\)

**Differentiation of Natural Logarithm**:

Differentiate \(j(x) = \ln(x)\).

Using the rule \(\frac{d}{dx} \ln(x) = \frac{1}{x}\):

\(j'(x) = \frac{1}{x}\)

These examples cover the basic differentiation rules and their application to different types of functions.

## Conclusion

Differentiation provides insight into the behavior of functions, allowing us to find maximum and minimum values, analyze rates of change, and solve a myriad of practical problems in various disciplines. As we delve deeper into calculus, we’ll find that differentiation is a powerful tool that underpins many advanced concepts and techniques.