Slide a second point towards the first: The derivative is the gradient of a chord whose two ends slide together until it becomes a tangent.
From chord to tangent
Pick a point
Pick a point on a curve.
Pick a second point
Pick a second point a little distance h to the right. The straight line joining them (a chord) has gradient (rise)/(run) = (f(x+h) − f(x))/h.
Slide it back
Let the second point slide back towards the first: h shrinks towards 0.
Tangent
The chord swings round until it becomes the tangent. Its gradient is the derivative f'(x).
Chord gradient, then let h → 0.
Why we can't just set h = 0: If you put h = 0 straight away you get 0/0, which is meaningless. The trick is to simplify the fraction first (the h's cancel), and only then let h → 0 in what's left. The limit tells us the value the gradient approaches, even though we never actually divide by zero.
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Expand, cancel the lone h, then let h → 0: Differentiating from first principles always uses the same four moves.
The four moves
Expand
Write f(x+h) and expand it.
Form the quotient
Subtract f(x) and form (f(x+h) − f(x))/h.
Cancel
Cancel the single h on the bottom (every top term has an h to give it).
Let h → 0
Let h → 0 — every leftover term still containing h disappears.
Expand, subtract, cancel, h → 0.
IB-style question — differentiate x² from first principles
A student must show, using the limit definition, that the derivative of f(x) = x² is 2x.
Differentiate f(x) = x² from first principles.
Step by step
- Write down the definition.
- Expand the top: (x+h)² = x² + 2xh + h². The x² terms cancel.
- Every top term has an h, so cancel the h on the bottom.
- Now let h → 0: the leftover h vanishes.
Final answer
f'(x) = 2x — matching the power rule (bring down the 2, drop the power by 1).
IB-style question — differentiate x³ from first principles
Show, from first principles, that the derivative of g(x) = x³ is 3x².
Step by step
- Set up the definition.
- Expand (x+h)³ = x³ + 3x²h + 3xh² + h³; the x³ terms cancel.
- Cancel one h from every term.
- Let h → 0: the two terms with an h vanish.
Final answer
g'(x) = 3x² — again confirming the power rule.