Fundamentals of Differentiation

Understand rates of change, gradients of curves, and the power rule in A-Level Pure Mathematics.

Learning Objectives

Understand dy/dx as the instantaneous rate of change of y with respect to x
Apply the power rule d/dx(xⁿ) = n·xⁿ⁻¹
Find gradients of tangents and normals to curves at specific points
Locate turning points by setting dy/dx = 0

Concept Explanation

Differentiation is the cornerstone of calculus. It allows mathematicians, physicists, and engineers to calculate the precise rate at which a variable changes at any given instant.

Gradient of a Curve: Unlike straight lines with a constant gradient, the gradient of a curve changes continuously at every point.

Derivative Notation: The derivative of y with respect to x is written as dy/dx or f'(x). It gives a general formula for the slope at any x.

The Power Rule: To differentiate term xⁿ, multiply by the power n and reduce the original power by 1: d/dx(xⁿ) = n·xⁿ⁻¹.

Constants & Polynomials: The derivative of any constant (like 5 or -12) is 0. Differentiation works term-by-term on sums of powers.

Key Mathematical Formulas

\frac{d}{dx}\left( x^n \right) = n x^{n-1}

\frac{d}{dx}\left( a x^n \right) = a n x^{n-1}

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

Worked Example

Step-by-Step Solution

Interactive Walkthrough

Problem Statement:

Find the derivative dy/dx and the gradient of the curve at x = 3 for:

y = 2x^3 - 5x^2 + 4x - 7

Step 1: Differentiate term by term using the Power Rule

Multiply each term coefficient by its power and decrease the exponent by 1.

\frac{dy}{dx} = 2(3x^2) - 5(2x) + 4(1x^0) - 0

Final Answer:

\frac{dy}{dx} = 6x^2 - 10x + 4, \quad \text{Gradient at } x=3 \text{ is } 28

Try It Yourself

What is the derivative dy/dx of the curve y = 4x³ - 6x² + 9?

y = 4x^3 - 6x^2 + 9

Common Student Mistakes

❌ Common Pitfall: Keeping the constant term when differentiating

✅ Correction: Constants have a rate of change of zero, so d/dx(c) = 0. Do not write +9 down in dy/dx.

❌ Common Pitfall: Confusing powers with negative exponents

✅ Correction: For 1/x = x⁻¹, reducing power by 1 gives -1 - 1 = -2, so d/dx(x⁻¹) = -x⁻² = -1/x².

UK Exam Advice & Tips

Before differentiating terms with roots or fractions like 3/√x, convert them to index notation first: 3x^(-1/2).
Stationary points (turning points) always occur where dy/dx = 0.

Lesson FAQ

What is the difference between a tangent and a normal?

A tangent is a line that touches the curve at a point and shares the exact same gradient (m_t = dy/dx). A normal is a line perpendicular to the tangent at that point, with gradient m_n = -1/m_t.

Ready to test your knowledge?

Solve interactive exam-style questions on Fundamentals of Differentiation with live scoring.

Start Practice Session