Calculus – Differentiation

Calculus- Differentiation

Lesson Objectives

By the end of this lesson, students should be able to:

Understand the concept of differentiation and its geometric interpretation.
Differentiate basic algebraic, trigonometric, exponential, and logarithmic functions.
Apply differentiation to solve real-life problems involving rates of change and tangents.

Lesson Introduction

Differentiation is a fundamental concept in calculus that deals with the rate at which quantities change. It allows us to find slopes of curves, velocities, accelerations, and more. In this lesson, we explore rules, techniques, and applications of differentiation.

Lesson Content

Definition of Differentiation

The derivative of a function $f(x)$ at a point $x$ is defined as:

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

Rules of Differentiation

Constant Rule: $\frac{d}{dx}(c) = 0$
Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
Sum Rule: $\frac{d}{dx}(f + g) = f' + g'$
Product Rule: $\frac{d}{dx}(fg) = f'g + fg'$
Quotient Rule: $\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}$
Chain Rule: $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$

Differentiation of Trigonometric Functions

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

Examples

Example 1:
Differentiate

f(x) = 3x^2 + 5x - 7

.
Solution:

f'(x) = 6x + 5

Example 2:
Differentiate

g(x) = \sin x + x^2

.
Solution:

g'(x) = \cos x + 2x

Example 3:
Differentiate

y = x^3 - 2x^2 + x

.
Solution:

y' = 3x^2 - 4x + 1

Example 4:
Differentiate

y = \frac{2x^2 + 1}{x}

.
Solution: Rewrite as

2x + \frac{1}{x}

, so

y' = 2 - x^{-2}

Example 5:
Differentiate

f(x) = e^x + \ln x

.
Solution:

f'(x) = e^x + \frac{1}{x}

Example 6:
Differentiate

y = x^2\sin x

using the product rule.
Solution:

y' = 2x\sin x + x^2\cos x

Example 7:
Differentiate

y = \frac{x}{x+1}

using the quotient rule.
Solution:

y' = \frac{(1)(x+1) - x(1)}{(x+1)^2} = \frac{1}{(x+1)^2}

Example 8:
Differentiate

f(x) = \ln(x^2 + 1)

.
Solution: Use chain rule:

f'(x) = \frac{2x}{x^2 + 1}

Example 9:
Find

\frac{d}{dx}(\tan^2 x)

.
Solution:

2\tan x \cdot \sec^2 x

Example 10:
Differentiate

y = \sqrt{x^2 + 4}

.
Solution:

y' = \frac{x}{\sqrt{x^2 + 4}}

Interactive Graph

Exercises

[WAEC] Differentiate $f(x) = x^3 + 4x^2 - x + 1$ [Past Question]
Find $\frac{d}{dx}(\cos x + x^2)$
[NECO] Differentiate $f(x) = \frac{3x + 2}{x - 1}$ [Past Question]
Find the derivative of $y = x^2 \cos x$
Differentiate $y = \ln(x^2 + 3x)$
[WASSCE] Find $\frac{d}{dx}(e^{2x} + \ln x)$ [Past Question]
Find the slope of $y = \sqrt{x^2 + 1}$ at $x = 2$
Differentiate $y = \tan x + x^3$
Find $\frac{d}{dx}(\sec x)$
[WAEC] Differentiate $y = \frac{x^2 + 1}{x^2 - 1}$ [Past Question]

Conclusion/Recap

In this lesson, we covered the fundamental rules and techniques of differentiation. We learned how to find derivatives of various types of functions and saw practical examples including algebraic, trigonometric, exponential, and logarithmic forms. You should now be confident in applying these rules to solve calculus problems.