A Level Maths Differentiation Explained: Step-by-Step Guide

Differentiation is a fundamental concept in A Level Mathematics. This step-by-step guide explains what differentiation is, how to differentiate different types of functions, and how to apply differentiation to solve problems, with clear explanations and worked examples.

What is Differentiation?

Differentiation is the process of finding the derivative of a function. The derivative tells us:

• The rate of change of a function
• The gradient of a curve at any point
• How fast something is changing

Notation:

• dy/dx (Leibniz notation)
• f’(x) (prime notation)
• y’ (shorthand)

Basic Differentiation Rules

Power Rule

• If y = xⁿ, then dy/dx = nxⁿ⁻¹
• Multiply by power, reduce power by 1
• Works for any real number n

Examples:

• y = x³ → dy/dx = 3x²
• y = x⁵ → dy/dx = 5x⁴
• y = x⁻² → dy/dx = -2x⁻³

Constant Rule

• If y = c (constant), then dy/dx = 0
• Constants differentiate to zero

Examples:

• y = 5 → dy/dx = 0
• y = -3 → dy/dx = 0

Sum/Difference Rule

• Differentiate each term separately
• d/dx[f(x) ± g(x)] = f’(x) ± g’(x)

Example:

• y = x³ + 2x² - 5x + 3
• dy/dx = 3x² + 4x - 5

Advanced Differentiation Rules

Product Rule

• If y = uv, then dy/dx = u’v + uv’
• Differentiate first × second + first × differentiate second

Example:

• y = x²(3x + 1)
• Let u = x², v = 3x + 1
• u’ = 2x, v’ = 3
• dy/dx = 2x(3x + 1) + x²(3) = 6x² + 2x + 3x² = 9x² + 2x

Quotient Rule

• If y = u/v, then dy/dx = (u’v - uv’) / v²
• Differentiate top × bottom - top × differentiate bottom, all over bottom squared

Example:

• y = (x² + 1) / (x + 2)
• Let u = x² + 1, v = x + 2
• u’ = 2x, v’ = 1
• dy/dx = [2x(x + 2) - (x² + 1)(1)] / (x + 2)²

Chain Rule

• If y = f(g(x)), then dy/dx = f’(g(x)) × g’(x)
• Differentiate outer function, multiply by derivative of inner function

Example:

• y = (3x + 2)⁵
• Outer function: u⁵, derivative: 5u⁴
• Inner function: 3x + 2, derivative: 3
• dy/dx = 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴

Applications of Differentiation

Finding Gradients

• dy/dx gives gradient at any point
• Substitute x-value to find gradient
• Positive gradient: increasing
• Negative gradient: decreasing

Stationary Points

• Where dy/dx = 0
• Maximum: gradient changes from + to -
• Minimum: gradient changes from - to +
• Points of inflection: gradient doesn’t change sign

Tangents and Normals

• Tangent: line touching curve at a point
• Gradient of tangent = dy/dx at that point
• Normal: perpendicular to tangent
• Gradient of normal = -1 / (gradient of tangent)

Optimization Problems

• Find maximum or minimum values
• Set up function to optimize
• Differentiate and set = 0
• Check if maximum or minimum

Step-by-Step Problem Solving

Problem 1: Find the derivative of y = 3x⁴ - 2x² + 5x - 1

1. Identify each term: 3x⁴, -2x², 5x, -1
2. Apply power rule to each: 12x³, -4x, 5, 0
3. Combine: dy/dx = 12x³ - 4x + 5

Problem 2: Find the gradient of y = x³ at x = 2

1. Differentiate: dy/dx = 3x²
2. Substitute x = 2: gradient = 3(2)² = 12

Problem 3: Find stationary points of y = x³ - 3x

1. Differentiate: dy/dx = 3x² - 3
2. Set = 0: 3x² - 3 = 0, so x² = 1, x = ±1
3. Find y-values: y(1) = -2, y(-1) = 2
4. Stationary points: (1, -2) and (-1, 2)

Common Mistakes to Avoid

1. Forgetting to Reduce Power

• Always reduce power by 1
• Check: power should be one less

2. Product Rule Errors

• Remember: u’v + uv’ (not u’v’)
• Don’t forget either term

3. Chain Rule Mistakes

• Don’t forget to multiply by inner derivative
• Work step by step

4. Sign Errors

• Be careful with negative signs
• Check your work

Related Resources

• A Level Maths Integration Explained
• A Level Maths Past Papers
• A Level Maths Tutor

---

Understanding differentiation is essential for A Level Maths. Practice regularly and seek help when needed to master this fundamental concept.