Differentiating powers
Differentiation finds an instantaneous rate of change.
For y = kxⁿ, dy/dx = knxⁿ⁻¹ wherever the function and derivative are defined. Differentiate a sum term by term. A constant differentiates to zero. Expand products first when a product rule is outside the course scope.
A derivative is a function, not just a number. Substitute the required input only after differentiating. Negative powers require x ≠ 0, and fractional powers may impose further restrictions. Preserve those conditions.
Worked example
Differentiate y = 3x³ − 5x² + 7.
- Differentiate 3x³ to get 9x².
- Differentiate −5x² to get −10x.
- The constant 7 contributes zero.
Answer: dy/dx = 9x² − 10x
Revise first: Fractional and negative indices.
Practise differentiating powers
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 4.3: Differentiating integer powers
- 9FM0 · core · Assumed A-level Mathematics knowledge: Differentiating integer powers
- 7367 · core · Assumed A-level Mathematics knowledge: Differentiating integer powers
- H245 · core · Assumed A-level Mathematics knowledge: Differentiating integer powers
- H645 · core · Assumed A-level Mathematics knowledge: Differentiating integer powers
- 6993 · core · CA1: Differentiating integer powers
- 4PM1 · core · 9A: Differentiating integer powers
- 8365 · core · 4.3: Gradient from a negative integer power
- 9FM0 · core · Assumed A-level Mathematics knowledge: Gradient from a negative integer power
- 7367 · core · Assumed A-level Mathematics knowledge: Gradient from a negative integer power
- H245 · core · Assumed A-level Mathematics knowledge: Gradient from a negative integer power
- H645 · core · Assumed A-level Mathematics knowledge: Gradient from a negative integer power
- 4PM1 · core · 9A: Gradient from a negative integer power
Next practice: Tangents and normals, Stationary points.