Higher Maths worked question

Find a gradient at a point by differentiation

A fixed, reviewed differentiating powers question with a complete worked solution and method explanation.

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Question

For y = x³ − 4x, find the gradient of the curve at x = 2.

Show the worked solution

Worked solution

  1. dy/dx = 3x² − 4

    Differentiate each term using the power rule.

  2. At x = 2: dy/dx = 3(2)² − 4

    The derivative gives the gradient at any value of x.

  3. dy/dx = 12 − 4 = 8

Final answer: The gradient is 8.

Why this method works

Differentiate the function before substituting the x-coordinate. Substituting into y would find a point on the curve, not its gradient.

Check your work

At x = 2 the positive 3x² term is larger than 4, so a positive gradient is reasonable

Common mistakes

  • Substituting x = 2 before differentiating
  • Differentiating −4x as −4x instead of −4.
  • Calculating the y-coordinate instead of the gradient.