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Differentiation from first principles

The gradient of a curve at a point is the limit of the gradients of chords that get shorter and shorter. This is where the rules of differentiation come from.

Chords

Take P at x and Q at x + h on y = f(x). The gradient of chord PQ is

(f(x + h) - f(x)) / h.

The limit

As h gets closer to 0, Q slides towards P and the chord becomes the tangent. So

f'(x) = the limit as h → 0 of (f(x + h) - f(x)) / h.

For f(x) = x2: ((x + h)2 - x2)/h = (2xh + h2)/h = 2x + h, which tends to 2x.

Meaning

dy/dx or f'(x) is the gradient of the curve, and also the rate of change of y with respect to x. If V is a volume and t is time, dV/dt is how fast the volume changes.

Worked example

Prove from first principles that the derivative of x3 is 3x2.

  1. (x + h)3 - x3 = 3x2h + 3xh2 + h3
  2. Divide by h: 3x2 + 3xh + h2
  3. As h → 0, the terms with h vanish

Answer: f'(x) = 3x2

Key idea: f'(x) = limit as h → 0 of (f(x + h) - f(x))/h. The derivative is both a gradient and a rate of change.

Check yourself

  1. Find the gradient of the chord of y = x2 from x = 3 to x = 3.1.
    Show answer

    6.1. (9.61 - 9)/0.1 = 0.61/0.1 = 6.1.

  2. For f(x) = x2, what does (f(x + h) - f(x)) / h simplify to?
    • 2x
    • 2xh + h2
    • 2x + h
    • x2 + h
    Show answer

    2x + h. (x2 + 2xh + h2 - x2)/h = (2xh + h2)/h = 2x + h.

  3. What is (x + h)3 - x3?
    • 3x2h + 3xh2 + h3
    • h3
    • 3xh + h3
    • 3x2h + h3
    Show answer

    3x2h + 3xh2 + h3. (x + h)3 = x3 + 3x2h + 3xh2 + h3. Subtract x3.

  4. The chord gradient of y = x3 between x = 2 and x = 2 + h is 12 + 6h + h2. What is the gradient of the curve at x = 2?
    Show answer

    12. Let h → 0: the gradient is 12. Check: 3x2 = 12 at x = 2.

  5. What does f'(3) represent?
    • The value of y when x = 3
    • The gradient of the curve y = f(x) at x = 3
    • The area under the curve up to x = 3
    • The gradient of the chord from x = 0 to x = 3
    Show answer

    The gradient of the curve y = f(x) at x = 3. f'(3) is the gradient of the tangent at x = 3.

  6. The volume of a balloon is V = 4t2 cm3 after t seconds. Find dV/dt when t = 3.
    Show answer

    24 cm3/s. dV/dt = 8t = 24 cm3 per second.

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Last updated 7 November 2026. Spotted a mistake? Tell us.