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This lesson covers the formal definition of the derivative using limits, as required by the Edexcel A-Level Mathematics specification (9MA0). You need to understand the concept of differentiation from first principles and be able to apply it to polynomials, sin x, cos x, and eˣ.
The derivative of a function f(x) is defined as:
f'(x) = lim[h→0] (f(x + h) - f(x)) / h
This is called differentiation from first principles. The expression (f(x + h) - f(x)) / h is the gradient of the chord between the points (x, f(x)) and (x + h, f(x + h)). As h → 0, the chord becomes the tangent, and the gradient of the chord becomes the gradient of the curve.
| Term | Meaning |
|---|---|
| f(x + h) - f(x) | The change in y (the rise) |
| h | The change in x (the run) |
| (f(x + h) - f(x)) / h | The gradient of the chord (average rate of change) |
| lim[h→0] | Take the limit as h approaches zero |
| f'(x) | The gradient of the tangent (instantaneous rate of change) |
f'(x) = lim[h→0] ((x + h)² - x²) / h
Expand (x + h)² = x² + 2xh + h²:
= lim[h→0] (x² + 2xh + h² - x²) / h = lim[h→0] (2xh + h²) / h = lim[h→0] (2x + h) [divide numerator and denominator by h] = 2x + 0 = 2x
So d/dx(x²) = 2x. ✓
f'(x) = lim[h→0] ((x + h)³ - x³) / h
Expand (x + h)³ = x³ + 3x²h + 3xh² + h³:
= lim[h→0] (x³ + 3x²h + 3xh² + h³ - x³) / h = lim[h→0] (3x²h + 3xh² + h³) / h = lim[h→0] (3x² + 3xh + h²) = 3x² + 0 + 0 = 3x²
So d/dx(x³) = 3x². ✓
f'(x) = lim[h→0] (5(x + h) + 3 - (5x + 3)) / h = lim[h→0] (5x + 5h + 3 - 5x - 3) / h = lim[h→0] 5h / h = lim[h→0] 5 = 5
So d/dx(5x + 3) = 5. A linear function has a constant gradient. ✓
Exam Tip: In an exam, the first principles question will tell you which function to differentiate. Show every algebraic step, including the expansion, simplification, cancelling of h, and then substituting h = 0. Do not skip steps.
To differentiate sin x from first principles, you need the limit result:
lim[h→0] sin h / h = 1 (where h is in radians)
and:
lim[h→0] (cos h - 1) / h = 0
f'(x) = lim[h→0] (sin(x + h) - sin x) / h
Use the addition formula: sin(x + h) = sin x cos h + cos x sin h
= lim[h→0] (sin x cos h + cos x sin h - sin x) / h = lim[h→0] [sin x (cos h - 1) + cos x sin h] / h = lim[h→0] [sin x × (cos h - 1)/h + cos x × sin h / h] = sin x × 0 + cos x × 1 = cos x
So d/dx(sin x) = cos x. ✓
| Limit | Value | Condition |
|---|---|---|
| lim[h→0] sin h / h | 1 | h must be in radians |
| lim[h→0] (cos h - 1) / h | 0 | h must be in radians |
| lim[h→0] (eʰ - 1) / h | 1 | This defines the number e |
Exam Tip: These limit results are given or expected knowledge at A-Level. You do not need to prove them, but you must use them correctly.
f'(x) = lim[h→0] (cos(x + h) - cos x) / h
Use the addition formula: cos(x + h) = cos x cos h - sin x sin h
= lim[h→0] (cos x cos h - sin x sin h - cos x) / h = lim[h→0] [cos x(cos h - 1) - sin x sin h] / h = cos x × 0 - sin x × 1 = -sin x
So d/dx(cos x) = -sin x. ✓
f'(x) = lim[h→0] (e^(x+h) - eˣ) / h
Factor out eˣ from the numerator: e^(x+h) = eˣ × eʰ
= lim[h→0] (eˣ eʰ - eˣ) / h = lim[h→0] eˣ(eʰ - 1) / h = eˣ × lim[h→0] (eʰ - 1) / h = eˣ × 1 = eˣ
So d/dx(eˣ) = eˣ. ✓
This is the defining property of e: it is the unique number for which lim[h→0] (eʰ - 1)/h = 1.
There are several equivalent notations for the derivative:
| Notation | Meaning |
|---|---|
| f'(x) | The derivative of f(x) — Lagrange notation |
| dy/dx | The derivative of y with respect to x — Leibniz notation |
| d/dx[f(x)] | Differentiate f(x) with respect to x — operator notation |
All three are used interchangeably at A-Level. Leibniz notation (dy/dx) is particularly useful for the chain rule and related rates.
First principles questions typically ask you to:
f'(x) = lim[h→0] [3(x + h)² + 2(x + h) - (3x² + 2x)] / h
Expand: 3(x² + 2xh + h²) + 2x + 2h - 3x² - 2x = 3x² + 6xh + 3h² + 2x + 2h - 3x² - 2x = 6xh + 3h² + 2h
Divide by h: 6x + 3h + 2
Take limit as h → 0: 6x + 2
Check: d/dx(3x² + 2x) = 6x + 2. ✓
| Mistake | Correction |
|---|---|
| Forgetting lim[h→0] in working | Always write lim[h→0] until you actually substitute h = 0 |
| Dividing by h before simplifying the numerator | Simplify the numerator first so h cancels cleanly |
| Using degrees instead of radians for trig | First principles for sin x and cos x only works in radians |
| Writing dy/dx = (f(x + h) - f(x)) / h (without the limit) | The derivative is the LIMIT of this expression, not the expression itself |
| Confusing f(x + h) with f(x) + h | f(x + h) means substitute (x + h) into f. For example, if f(x) = x², then f(x + h) = (x + h)² |