Higher Maths worked question

Find a gradient from a polynomial derivative

Practise differentiating a polynomial and finding the gradient at a given point, then check your method with a full worked solution.

Standard3 marksCalculator optional

Question

For y = 2x3 − 5x2 + 4, find the gradient of the curve at x = 3.

Show the worked solution

Worked solution

  1. dy/dx = 6x2 − 10x

    Apply the power rule to 2x3 and −5x2; the constant 4 differentiates to zero

  2. At x = 3: dy/dx = 6(3)2 − 10(3)

    The derivative gives the gradient of the curve at the specified x-coordinate.

  3. dy/dx = 54 − 30 = 24

    Evaluate the squared term before subtracting the linear term.

Final answer: 24

Why this method works

A curve's gradient comes from its derivative, not from the value of y. Differentiate every term first, then evaluate the resulting gradient function at x = 3.

Check your work

At x = 3, the positive contribution 6x2 is 54 and the negative contribution 10x is 30, so a positive gradient of 24 is consistent with the derivative.

Common mistakes

  • Substituting x = 3 into the original function instead of the derivative
  • Differentiating −5x2 as −5x
  • Keeping the constant 4 in the derivative.