Basic Differentiation

This lesson covers the power rule, the sum/difference rule, and applications to tangents and normals, as required by the Edexcel A-Level Mathematics specification (9MA0). These are the foundational differentiation skills you will use throughout the course.

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The Power Rule

The most important differentiation rule:

If y = xⁿ, then dy/dx = nxⁿ⁻¹

This works for any real value of n — positive, negative, fractional, or zero.

y	dy/dx
x²	2x
x³	3x²
x⁵	5x⁴
x	1 (since x = x¹)
x⁰ = 1	0 (derivative of a constant)
x⁻¹ = 1/x	-x⁻² = -1/x²
x⁻² = 1/x²	-2x⁻³ = -2/x³
x^(1/2) = √x	(1/2)x^(-1/2) = 1/(2√x)
x^(3/2)	(3/2)x^(1/2) = (3/2)√x

Constant Multiple Rule

If y = kxⁿ (where k is a constant), then:

dy/dx = knxⁿ⁻¹

"The constant stays, differentiate the power as normal."

Worked Example 1

Differentiate: (a) y = 4x³ (b) y = 7x (c) y = -2x⁵ (d) y = (1/3)x⁶