[ADD MATHS] Form 5 Chapter 2 - Differentiation Part 2 (Basic Differentiation) | KSSM

Introduction

This tutorial is designed to help you understand the basics of differentiation as covered in the ADD MATHS Form 5 Chapter 2 - Differentiation Part 2. Differentiation is a fundamental concept in calculus, allowing us to determine the rate at which a function is changing. This guide will break down the essential steps and techniques involved in basic differentiation, making it easier for you to grasp and apply these concepts in your studies.

Step 1: Understanding Basic Differentiation

To start with differentiation, it’s important to understand what it entails. Differentiation is the process of calculating the derivative of a function, which tells us the slope of the function at any given point.

• Definition of Derivative: 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} ]

• Notation: The derivative can be denoted as f'(x) or (\frac{dy}{dx}) where y = f(x).

Practical Tips

• Always remember that differentiation is focused on instantaneous rates of change.
• Familiarize yourself with common derivatives, such as:
  • ( \frac{d}{dx}(x^n) = nx^{n-1} )
  • ( \frac{d}{dx}(\sin x) = \cos x )
  • ( \frac{d}{dx}(\cos x) = -\sin x )

Step 2: Applying the Power Rule

The power rule is one of the most useful techniques in differentiation. It provides an efficient way to find the derivative of polynomial functions.

• Power Rule: If ( f(x) = x^n ), then: [ f'(x) = nx^{n-1} ]

Steps to Apply the Power Rule

1. Identify the exponent ( n ) in the term ( x^n ).
2. Multiply the term by the exponent.
3. Decrease the exponent by one.

Example

• Differentiate ( f(x) = x^3 ):
  • ( f'(x) = 3x^{3-1} = 3x^2 )

Step 3: Differentiating Common Functions

In addition to using the power rule, certain functions have standard derivatives that you should memorize.

Common Derivatives

• For constant functions:
  • If ( f(x) = c ), then ( f'(x) = 0 )
• For exponential functions:
  • If ( f(x) = e^x ), then ( f'(x) = e^x )
• For logarithmic functions:
  • If ( f(x) = \ln(x) ), then ( f'(x) = \frac{1}{x} )

Practical Advice

• Create a flashcard set for these common derivatives for quick reference.
• Practice differentiating a variety of functions to build confidence.

Step 4: Using the Product and Quotient Rules

When dealing with products or quotients of functions, you’ll need to apply the product and quotient rules.

Product Rule

If ( f(x) = g(x) \cdot h(x) ), then: [ f'(x) = g'(x)h(x) + g(x)h'(x) ]

Quotient Rule

If ( f(x) = \frac{g(x)}{h(x)} ), then: [ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} ]

Example Using Product Rule

• Differentiate ( f(x) = x^2 \sin x ):
  • Let ( g(x) = x^2 ) and ( h(x) = \sin x )
  • Then apply the product rule: [ f'(x) = (2x)\sin x + (x^2)(\cos x) ]

Conclusion

In this tutorial, we covered the basics of differentiation, including the power rule, common derivatives, and the product and quotient rules. Mastering these concepts will enhance your understanding of calculus and prepare you for more advanced topics.

Next Steps

• Practice differentiating a variety of functions.
• Explore applications of derivatives in real-world scenarios, such as physics and economics.
• Consider joining study groups or online forums for additional support and resources.