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Extended Only Algebra and Graphs

Finding Gradients for IGCSE Maths

Using differentiation to find the gradient at a specific point on a curve. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding

What You Need to Know

Using differentiation to find the gradient at a specific point on a curve. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding finding gradients is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Finding Gradients

To find the gradient of a curve y = f(x) at a specific point, differentiate to get dy/dx, then substitute the x-coordinate of the point. The result is the gradient of the tangent at that point. Unlike a straight line (constant gradient), a curve has a different gradient at every point — this is the key idea of calculus. Tested in IGCSE Extended Paper 4, often as the first step in tangent/normal equations or stationary point problems.

Step-by-Step Method

  1. 1

    Differentiate the function

    Find dy/dx by applying the power rule to each term.

  2. 2

    Substitute the x-coordinate

    Replace x in dy/dx with the given x-value. This gives a number — the gradient at that specific point.

  3. 3

    State the gradient clearly

    Write 'When x = a, dy/dx = m' or 'Gradient at (a, b) = m'.

  4. 4

    Find the y-coordinate if needed

    Substitute x = a into the original equation y = f(x) to find the y-coordinate of the point.

  5. 5

    Interpret the gradient

    Positive gradient → curve is increasing. Negative → decreasing. Zero → stationary point.

Worked Example

Question

y = x³ − 4x + 2. Find the gradient of the curve at the point where x = 2.

Solution

Step 1: Differentiate. dy/dx = 3x² − 4 Step 2: Substitute x = 2. dy/dx = 3(2)² − 4 = 3(4) − 4 = 12 − 4 = 8 Step 3: Find the y-coordinate at x = 2. y = (2)³ − 4(2) + 2 = 8 − 8 + 2 = 2 Answer: Gradient = 8, at the point (2, 2).

Exam Tips for Finding Gradients

  • Gradient at a point = substitute x into dy/dx — not into y itself.
  • Show the substitution step explicitly: write dy/dx = ... then = ... after substituting x.
  • For 'find the point with gradient k', set dy/dx = k and solve for x — this is a common exam question type.
  • A negative gradient at a point means the curve is sloping downward at that x-value.

Practice Questions

Q1: y = 2x³ − 5x². Find the gradient at x = −1.

Show hint

dy/dx = 6x²−10x. At x=−1: 6(1)−10(−1) = 6+10 = 16.

Q2: Find the coordinates of the point on y = x² − 6x + 11 where the gradient is 4.

Show hint

dy/dx = 2x−6 = 4 → 2x=10 → x=5. y = 25−30+11 = 6. Point: (5, 6).

Q3: y = x³ + 3x² − 9x. At what x-values is the gradient equal to 0? What do these points represent?

Show hint

dy/dx = 3x²+6x−9 = 3(x²+2x−3) = 3(x+3)(x−1) = 0. x=−3 or x=1. These are stationary points.

Frequently Asked Questions

What is finding gradients in IGCSE Maths?

Using differentiation to find the gradient at a specific point on a curve.

Is finding gradients in the Core or Extended syllabus?

Finding Gradients is part of the Extended only syllabus for IGCSE Mathematics 0580.

How do I revise finding gradients effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on finding gradients rather than trying to cover everything at once.

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