The Gradient Machine: Mastering Differentiation

By Joshua Ramirez, MSc·Updated April 18, 2026
A dramatically lit blackboard with calculus notation.

What is the single most important rule in Differentiation?

The Power Rule: If y = ax^n, then dy/dx = nax^(n-1). Bring the power down as a multiplier, then subtract 1 from the power. This single rule handles 80% of all differentiation questions. Example: y = 5x^3 gives dy/dx = 15x^2. For constants (like y = 7), dy/dx = 0 because constants have zero rate of change.

Table of Contents

Differentiation is the single most heavily tested topic in the entire CAIE Additional Mathematics syllabus. If you cannot differentiate fluently, you cannot do integration, you cannot find stationary points, and you cannot answer any of the Paper 2 application questions. This guide from our Ultimate Add Maths Guide builds your calculus instinct from scratch.

1. The Power Rule (The Foundation)

Every polynomial differentiation uses this one rule. Memorize it like breathing.

Rule: If y = ax^n, then dy/dx = n × a × x^(n-1)

Example 1: y = 4x^5 → dy/dx = 20x^4

Example 2: y = -2x^3 + 7x → dy/dx = -6x^2 + 7

Example 3: y = 3/x^2 = 3x^(-2) → dy/dx = -6x^(-3) = -6/x^3

💡 Tutor's Tip
Rewrite First! Before differentiating, always rewrite fractions and roots as negative/fractional powers. √x = x^(1/2). So dy/dx of √x = (1/2)x^(-1/2) = 1/(2√x). Students who skip this rewrite step get stuck immediately.

2. The Chain Rule (Composite Functions)

When you have a function inside another function (like y = (3x+1)^5), the Power Rule alone is not enough. You need the Chain Rule: dy/dx = dy/du × du/dx.

Worked Example: y = (3x + 1)^5

Step 1: Let u = 3x + 1. Then y = u^5.

Step 2: dy/du = 5u^4. du/dx = 3.

Step 3: dy/dx = 5u^4 × 3 = 15(3x+1)^4

3. Finding Stationary Points (Max/Min)

A stationary point is where the curve momentarily flattens out — the gradient is exactly zero. To find these turning points, set dy/dx = 0 and solve.

The Second Derivative Test

After finding the x-value where dy/dx = 0, differentiate again to get d²y/dx².

If d²y/dx² > 0 → MINIMUM (curve bends upward like a bowl)

If d²y/dx² < 0 → MAXIMUM (curve bends downward like a hill)

💡 Tutor's Tip
Mnemonic: "Negative = Naughty Hill" If d²y/dx² is negative, the curve is curving downward — it's going down, like a "naughty hill" you'd slide down. That means it's a maximum.

4. Equations of Tangents and Normals

A tangent line just touches the curve at a point. A normal line is perpendicular to the tangent at that same point.

The Process

1. Differentiate to get dy/dx. 2. Substitute the x-value to find the gradient m. 3. Use y - y1 = m(x - x1) for the tangent. 4. For the normal, use gradient = -1/m (the negative reciprocal).

Joshua Ramirez📋 From the Desk of Joshua Ramirez
The "Show That" Trap:In "Show that the gradient at x=2 is 7" questions, you MUST show every single step of working. If your answer jumps straight from dy/dx = 3x^2 - 5 to "= 7", the examiner cannot give you the marks even though you are correct. Explicitly write: dy/dx at x=2 = 3(2)^2 - 5 = 12 - 5 = 7.

Frequently Asked Questions

What does dy/dx actually mean?▼
The rate of change of y with respect to x — geometrically, it is the exact gradient of the tangent to the curve at any single point.
What is the Power Rule?▼
If y = ax^n, then dy/dx = nax^(n-1). Multiply the coefficient by the power, then reduce the power by 1.
How do I find a stationary point?▼
Set dy/dx = 0 and solve for x. This locates the point(s) where the gradient is zero (flat tangent).
How do I determine if it is a maximum or minimum?▼
Use the second derivative test. If d2y/dx2 > 0, it is a minimum. If d2y/dx2 < 0, it is a maximum.

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