Differentiation is one of the most important topics in A Math Discover the best math tuition Singapore for targeted exam preparation and concept mastery.. This guide breaks it down into simple, manageable steps.
What Is Differentiation?
Differentiation finds the rate of change of a function. It tells us how quickly something is changing at any point.
In simple terms: If y = f(x), the derivative dy/dx tells us how y changes when x changes.
Basic Rules of Differentiation
- Power Rule
- If y = x^n, then dy/dx = nx^(n-1)
Examples:
- y = x^3, dy/dx = 3x^2
- – y = x^5, dy/dx = 5x^4
- – y = x, dy/dx = 1
2. Constant Multiple Rule
If y = kx^n, then dy/dx = knx^(n-1)
Examples:
- y = 5x^2, dy/dx = 10x
- – y = 3x^4, dy/dx = 12x^3
3. Sum/Difference Rule
Differentiate each term separately.
Examples:
- y = x^3 + 2x^2, dy/dx = 3x^2 + 4x
- – y = 4x^3 – 2x, dy/dx = 12x^2 – 2
4. Constant Rule
The derivative of a constant is 0.
Examples:
- y = 5, dy/dx = 0
- – y = x^2 + 7, dy/dx = 2x
Differentiating Different Functions
Polynomials:
Apply the power rule to each term.
Example: y = 2x^4 – 3x^2 + 5x – 1
dy/dx = 8x^3 – 6x + 5
Fractional Powers:
Same rule applies.
Example: y = x^(1/2) = √x
dy/dx = (1/2)x^(-1/2) = 1/(2√x)
Negative Powers:
Same rule applies.
Example: y = x^(-2) = 1/x^2
dy/dx = -2x^(-3) = -2/x^3
Chain Rule
Used for composite functions.
If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x)
In simpler form: dy/dx = (dy/du) × (du/dx)
Examples:
y = (2x + 1)^3
Let u = 2x + 1, so y = u^3
dy/du = 3u^2 = 3(2x + 1)^2
du/dx = 2
dy/dx = 3(2x + 1)^2 × 2 = 6(2x + 1)^2
Product Rule
Used when differentiating a product of two functions.
If y = u × v, then dy/dx = u(dv/dx) + v(du/dx)
Example: y = x^2 × (2x + 1)
u = x^2, du/dx = 2x
v = 2x + 1, dv/dx = 2
dy/dx = x^2(2) + (2x + 1)(2x)
dy/dx = 2x^2 + 4x^2 + 2x = 6x^2 + 2x
Quotient Rule
Used when differentiating a quotient of two functions.
If y = u/v, then dy/dx = [v(du/dx) – u(dv/dx)] / v^2
Example: y = x/(x + 1)
u = x, du/dx = 1
v = x + 1, dv/dx = 1
dy/dx = [(x + 1)(1) – x(1)] / (x + 1)^2
dy/dx = 1/(x + 1)^2
Applications of Differentiation
- Gradient of a Curve
- dy/dx gives the gradient at any point.
2. Tangent and Normal
- Tangent gradient = dy/dx at that point
- – Normal gradient = -1/(dy/dx)
3. Stationary Points
Where dy/dx = 0
- Maximum point: d^2y/dx^2 < 0
- – Minimum point: d^2y/dx^2 > 0
4. Rate of Change
Connected rates of change use chain rule.
Common Mistakes
Forgetting to reduce the power:
y = x^3, dy/dx ≠ 3x^3 (wrong!)
dy/dx = 3x^2 (correct)
Not applying chain rule:
y = (3x + 1)^2, dy/dx ≠ 2(3x + 1)
dy/dx = 2(3x + 1) × 3 = 6(3x + 1)
Mixing up product and chain rules:
Identify which rule to use before differentiating.
Practice Tips
Start with basic power rule questions.
Add chain rule gradually.
Practise product and quotient rules.
Mix all types for exam preparation.
How Ace Scorers Helps
Our A Math programme builds differentiation skills systematically:
- Clear concept introduction
- – Progressive practice
- – Exam-focused preparation
- – Individual attention
Contact us for A Math tuition.
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