First principles

Definition

Differentiation from first principles uses f′(x) = lim(h→0) [f(x + h) − f(x)]/h. The limit notation must be kept on every line until h has been cancelled, otherwise method marks are lost.

Worked example

For f(x) = x²: f′(x) = lim(h→0)[(x + h)² − x²]/h = lim(h→0)(2x + h) = 2x.

For f(x) = 3x²: f(x + h) = 3x² + 6xh + 3h², so [f(x + h) − f(x)]/h = (6xh + 3h²)/h = 6x + 3h, and letting h → 0 gives f′(x) = 6x.

Where it appears in the exam

Paper 1 calculus regularly includes a first principles question, usually on a quadratic function.

Step-by-step method

1. Find f(x + h) by replacing every x with (x + h).
2. Write f′(x) = lim(h→0) [f(x + h) − f(x)]/h.
3. Expand and simplify the numerator, factor out h and cancel it.
4. Let h → 0 to get the derivative.

Common mistake

Dropping the 'lim h→0' notation before h has been cancelled, which costs marks.

Related calculus terms

• Stationary point
• Point of inflection
• Optimisation

Frequently asked questions

What does first principles mean in Grade 12 Maths?

Differentiation from first principles uses f′(x) = lim(h→0) [f(x + h) − f(x)]/h. The limit notation must be kept on every line until h has been cancelled, otherwise method marks are lost.

Which NSC paper tests first principles?

First principles belongs to the Calculus section of CAPS Grade 12 Mathematics, which is examined in Paper 1.

How do I practise questions involving first principles?

Snap a question that uses first principles into the Snap&Learn AI solver, or work through the calculus topic guide and NSC past papers.