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Learn Index · Calculus Grade 11–12

Derivatives from first principles

f′(x) = lim (h→0) (f(x + h) − f(x)) / h.

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Building the derivative from scratch

Before using shortcuts, it’s important to see where derivatives come from. Finding a derivative from first principles uses the limit definition:

f′(x) = lim (h→0) [f(x + h) − f(x)] / h

From secant to tangent

The difference quotient [f(x + h) − f(x)]/h is the slope of the secant line through (x, f(x)) and (x + h, f(x + h)).

As h shrinks to 0, the second point slides toward the first, and the secant becomes the tangent line.

Example 1: f(x) = x²

  1. f(x + h) = x² + 2xh + h²
  2. f(x + h) − f(x) = 2xh + h²
  3. Divide by h: 2x + h
  4. Let h → 0: f′(x) = 2x

Example 2: f(x) = 3x² − 5x

  1. f(x + h) = 3(x + h)² − 5(x + h) = 3x² + 6xh + 3h² − 5x − 5h
  2. Subtract f(x): 6xh + 3h² − 5h
  3. Divide by h: 6x + 3h − 5
  4. h → 0: f′(x) = 6x − 5

Example 3: f(x) = x³

  1. (x + h)³ = x³ + 3x²h + 3xh² + h³
  2. Subtract: 3x²h + 3xh² + h³
  3. Divide: 3x² + 3xh + h²
  4. h → 0: f′(x) = 3x²

Example 4: f(x) = 1/x

  1. 1/(x + h) − 1/x = [x − (x + h)]/[x(x + h)] = −h/[x(x + h)]
  2. Divide by h: −1/[x(x + h)]
  3. h → 0: f′(x) = −1/x²

Example 5: f(x) = √x

  1. [√(x + h) − √x]/h × [√(x + h) + √x]/[√(x + h) + √x]
  2. = h/{h[√(x + h) + √x]} = 1/[√(x + h) + √x]
  3. h → 0: f′(x) = 1/(2√x)

At a specific point

f′(a) = lim (h→0) [f(a + h) − f(a)]/h

Example: f(x) = x², a = 4: [(4 + h)² − 16]/h = 8 + h → 8.

Strategy

  1. Write f(x + h) carefully (brackets!).
  2. Subtract f(x) and simplify: the terms without h should cancel.
  3. Factor out h and cancel it.
  4. Take the limit as h → 0.

If the terms without h don’t cancel in step 2, there’s an algebra mistake.

Worked examples

Example 1: f(x) = 5x + 2 → [5(x + h) + 2 − 5x − 2]/h = 5 → 5.

Example 2: f(x) = 4 → (4 − 4)/h = 0 → 0.

Example 3: f(x) = x² + x → 2x + 1.

Common mistakes

  • Writing f(x + h) as f(x) + h.
  • Dropping brackets when expanding.
  • Setting h = 0 before cancelling (gives 0/0).

Try it yourself

Use first principles:

  1. f(x) = 7x
  2. f(x) = x² − 3x
  3. f(x) = 2x²
  4. f′(1) for f(x) = x³

Answers: 1) 7 2) 2x − 3 3) 4x 4) 3

Quick recap

  • First principles: f′(x) = lim (h→0) [f(x + h) − f(x)]/h.
  • The difference quotient is a secant line slope; its limit is the tangent line slope.
  • Expand, simplify, cancel h, then take the limit.