Differentiation Basics

From limits to a shortcut

In introduction to calculus: rates of change we found the gradient of a curve the slow way — taking a chord and letting the gap h shrink to zero. Doing that every time would be exhausting. Differentiation is the shortcut: a set of rules that give the gradient instantly.

The result of differentiating is the gradient function, written dy/dx (read "dee y by dee x") or f'(x) (read "f dashed of x"). It is a new formula: feed in any x-value and it returns the gradient of the curve there.

The power rule

The single most useful rule is the power rule:

If y = xⁿ, then dy/dx = n · xⁿ⁻¹

In words: bring the power down to the front as a multiplier, then reduce the power by one.

Worked example 1: Differentiate y = x³.

1. The power is n = 3.
2. Bring it down: 3 × x. Reduce the power by 1: x³⁻¹ = x².
3. dy/dx = 3x².

Worked example 2: Differentiate y = x⁷.

1. Bring down the 7, reduce the power to 6.
2. dy/dx = 7x⁶.

Constant multipliers

If a term has a number in front, that multiplier stays and you differentiate the power as normal.

If y = a·xⁿ, then dy/dx = a·n·xⁿ⁻¹

Worked example 3: Differentiate y = 5x².

1. Keep the 5. Differentiate x²: 2x.
2. dy/dx = 5 × 2x = 10x.

Worked example 4: Differentiate y = 2x⁴.

1. Keep the 2. Differentiate x⁴: 4x³.
2. dy/dx = 2 × 4x³ = 8x³.

Two special cases: x and constants

• The term x is really x¹. By the power rule: 1 · x⁰ = 1 (since x⁰ = 1). So x differentiates to 1, and 5x differentiates to 5.
• A lone constant like 9 is a flat line with zero gradient, so a constant differentiates to 0.

Term	Derivative
x⁴	4x³
6x²	12x
x	1
3x	3
10	0

Differentiating a polynomial

A polynomial is a sum of terms, and you simply differentiate each term separately.

Worked example 5: Differentiate y = x³ + 4x² − 7x + 2.

1. x³ → 3x².
2. 4x² → 8x.
3. −7x → −7.
4. +2 → 0.
5. Collect: dy/dx = 3x² + 8x − 7.

Worked example 6: Differentiate y = 2x⁵ − 3x³ + x − 6.

1. 2x⁵ → 10x⁴.
2. −3x³ → −9x².
3. x → 1.
4. −6 → 0.
5. dy/dx = 10x⁴ − 9x² + 1.

If a function comes with brackets, expand it first using expanding brackets so that every term is a simple power of x before you differentiate.

Worked example 7: Differentiate y = x(x + 3).

1. Expand: y = x² + 3x.
2. Differentiate: dy/dx = 2x + 3.

Finding the gradient at a point

The derivative is a formula. To get the gradient at one specific point, substitute that x-value into dy/dx.

Worked example 8: For y = x² − 4x + 1, find the gradient at x = 5.

1. Differentiate: dy/dx = 2x − 4.
2. Substitute x = 5: 2(5) − 4 = 10 − 4 = 6.
3. The gradient at x = 5 is 6.

Worked example 9: For y = x³ − 6x, find where the gradient equals 0.

1. Differentiate: dy/dx = 3x² − 6.
2. Set equal to 0: 3x² − 6 = 0, so 3x² = 6, x² = 2, x = ±√2.
3. The curve is momentarily flat at x = √2 and x = −√2 — these are the turning points (peaks and troughs).

Solving dy/dx = 0 uses ordinary equation-solving — revise solving linear equations if rearranging feels shaky.

Negative and fractional powers

The power rule never changes. You just rewrite the function as a power first.

Worked example 10: Differentiate y = 1/x².

1. Rewrite: y = x⁻².
2. Power rule: bring down −2, reduce power by 1: −2 · x⁻³.
3. dy/dx = −2x⁻³ = −2/x³.

Worked example 11: Differentiate y = √x.

1. Rewrite: y = x^(1/2).
2. Bring down ½, reduce power by 1: ½ · x^(−1/2).
3. dy/dx = ½x^(−1/2) = 1/(2√x).

Where differentiation is used

Differentiation finds maximum and minimum values — the most profit, the least material, the highest point of a thrown ball — by solving dy/dx = 0. Physicists differentiate position to get velocity, and velocity to get acceleration. Engineers use it to find where stress on a beam is greatest. Economists differentiate cost and revenue to optimise. Whenever you need "the rate right now" or "the best value," differentiation is the tool.

Practice activity

1. Differentiate y = x⁶.
2. Differentiate y = 4x³ + 2x² − x.
3. Differentiate y = (x + 2)(x − 5) (expand first).
4. For y = x² − 6x + 4, find the gradient at x = 1.
5. For y = x³ − 12x, find the x-values where the gradient is 0.

Answers:

1. dy/dx = 6x⁵.
2. dy/dx = 12x² + 4x − 1.
3. Expand: x² − 3x − 10; differentiate → 2x − 3.
4. dy/dx = 2x − 6; at x = 1 → 2 − 6 = −4.
5. 3x² − 12 = 0 → x² = 4 → x = ±2.

Summary

Differentiation turns a curve y into its gradient function dy/dx. The key rule is the power rule: xⁿ → n·xⁿ⁻¹. Constant multipliers stay, x differentiates to 1, and a lone constant differentiates to 0. Differentiate polynomials term by term, expanding brackets first. Substitute an x-value into dy/dx for the gradient there, and solve dy/dx = 0 to find flat turning points. The same power rule even handles negative and fractional powers once you rewrite roots and reciprocals.