An introductory worksheet on differentiating simple functions from first principles using the definition of the derivative at x0 as the limit of the quotient f(x)-f(x0)/x-x0 at this point. Detailed solutions are included.

Master differentiating from first principles with this comprehensive, no-prep calculus maths worksheet pack featuring polynomial and rational functions. Designed specifically for A-Level Mathematics, Further Maths, and advanced secondary calculus lessons, this print-and-go resource pack contains 10 highly targeted practice problems. A complete, step-by-step worked solutions guide is included for every question to entirely eliminate your marking time and provide easy diagnostic scoring.

What is Differentiation from First Principles?
Differentiation from first principles is the foundational method in calculus used to find the exact rate of change (or gradient) of a function at any given point. Instead of relying on shortcut derivative rules, pupils must evaluate the limit of a difference quotient as the distance between two points approaches zero. Mastering this algebraic limiting process is a core requirement for secondary exam frameworks and provides the fundamental proof behind all calculus operations.

What is Included in This Resource Pack:
• Constant & Linear Questions (1–3) – Foundational entry-level practice finding the derivative at a specific point for simple functions like f(x) = 7 and f(x) = 3x + 2.
• Quadratic Expressions (4–6) – Core calculation problems requiring pupils to expand equations and factorise quadratics to resolve indeterminate forms.
• Cubic Functions (7–8) – Advanced algebraic challenges requiring the expansion and factorisation of cubed variables.
• Rational Functions (9–10) – High-level fractional equations requiring pupils to find a common denominator and manipulate algebraic fractions to evaluate limits successfully.
• Complete Step-by-Step Solutions Guide – A full, line-by-line worked math answers section showcasing every single algebraic phase, difference quotient expansion, and factorisation step for easy student self-correction or quick marking.

Why Secondary Maths Teachers Love This Pack:
• Zero Planning Required: Simply download the high-quality PDF files, print the ready-made worksheets, and hand them directly to your class.
• Scaffolded Learning Flow: Problems naturally build in difficulty—moving from simple constants to multi-term quadratics, cubics, and challenging rational expressions.
• Versatile Classroom Implementation: An ideal choice for independent homework packs, classroom maths starter activities, sixth-form revision folders, early finishers, or end-of-unit exam preparation.