Mathematics 0580 · Syllabus tracker
Differentiation
Differentiating axⁿ and sums of terms, gradients and tangents to curves, and turning points with maximum or minimum.
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Quick revision
Differentiation on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Differentiating axⁿ, gradients and tangents, and turning points with maximum or minimum.
1Differentiating
- The derivative dy/dx gives the gradient of the curve at any point.
- For axⁿ: multiply by the power, then take one off the power.
- A term in x differentiates to a number (7x → 7); a constant differentiates to 0.
- Differentiate a sum term by term.
Formulas
Worked example
Differentiate y = x4 − 3x2 + 7x − 2.
- x4 → 4x3.
- −3x2 → −6x.
- 7x → 7; −2 → 0.
dy/dx = 4x3 − 6x + 7
2Gradients and tangents
- Gradient at a point: differentiate, then substitute the x value.
- Where is the gradient k? Set dy/dx = k and solve.
- Equation of the tangent at (x₁, y₁): m = dy/dx there, then y − y₁ = m(x − x₁).
Formulas
Worked example
Find the equation of the tangent to y = x2 + 3x at x = 1.
- y = 1 + 3 = 4 at x = 1.
- dy/dx = 2x + 3 = 5 at x = 1.
- y − 4 = 5(x − 1).
y = 5x − 1
3Turning points
- At a turning (stationary) point the gradient is 0: solve dy/dx = 0.
- Find y by substituting x into the original equation.
- Maximum or minimum? Second derivative d²y/dx²: negative → maximum, positive → minimum. Or check the gradient just either side.
- Real-life problems: maximise area, profit, volume by finding where the derivative is 0.
How to do it
- Differentiate.
- Set dy/dx = 0 and solve for x.
- Substitute into y for the coordinates.
- Use d²y/dx² (or the gradient either side) to decide max or min.
Worked example
Find the turning points of y = x3 − 12x and say which is a maximum.
- dy/dx = 3x2 − 12 = 0 → x = ±2.
- x = 2: y = 8 − 24 = −16. x = −2: y = −8 + 24 = 16.
- d²y/dx² = 6x: at x = 2 it is 12 (minimum); at x = −2 it is −12 (maximum).
Maximum (−2, 16); minimum (2, −16)
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Differentiating 5x3 to 5x2 | Multiply by the power too: 15x2 |
| Leaving the constant in the derivative | Constants differentiate to 0 |
| Finding y at a turning point from dy/dx | Substitute into the original equation |
| 7x differentiates to 7x0 left as 7x | 7x → 7 |
★In the exam
- Write dy/dx = … on its own line before substituting.
- State the coordinates of turning points, not just x.
- Say clearly 'maximum' or 'minimum' with your reason.