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Calculus, Matrices & Vectors · SS 3

Differentiation: power rule

Find the gradient function of a curve.

  • If y = axn, then dydx = naxn − 1
  • The derivative of a constant is 0

What the letters mean

dydx
the derivative: the gradient at any point
a
the number in front
n
the power of x

Worked example

Find the gradient of y = 3x4 at x = 2.

  1. Step 1: Differentiate

    dydx = 4 × 3x3 = 12x3

  2. Step 2: Put x = 2

    12 × 23 = 12 × 8 = 96

Final answer: 96

Bring the power down to multiply, then reduce the power by 1.

See it

Play with this picture from the Rules of Differentiation lesson.

P

curve · tangent at P

y = x2 → dy/dx = 2x

  • x2 → 2 × 1x1 = 2x

P(1, 1): dy/dx = 2 × 1 = 2

P: x1
a1
power n2
b0
c0