Home » Differentiation

# Differentiation

## 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

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

## 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:

1. 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:

1. 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.

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept