Tangents and normals
The derivative gives a tangent’s gradient at the chosen point.
Read the lesson: Tangents and normals
Find the point by substituting into the original function, and find the gradient by substituting into the derivative. Use both in y − y₀ = m(x − x₀). For y = x² at x = 3, for example, the point is (3, 9) and the gradient is 6.
The normal is perpendicular to the tangent. Its gradient is −1/m when the tangent gradient m is nonzero. If the tangent is horizontal, the normal is vertical and has equation x = x₀.
Worked example
Find the tangent to y = x² + 1 at x = 2.
- The point is (2,5).
- dy/dx = 2x gives gradient 4 at x = 2.
- Write y − 5 = 4(x − 2).
Answer: y = 4x − 3
Revise first: Differentiating powers, Straight lines and gradients.
Practise tangents and normals
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 4.4: Tangents and normals
- 8365 · core · 4.4: Equation of a tangent
- 9FM0 · core · Assumed A-level Mathematics knowledge: Equation of a tangent
- 7367 · core · Assumed A-level Mathematics knowledge: Equation of a tangent
- H245 · core · Assumed A-level Mathematics knowledge: Equation of a tangent
- H645 · core · Assumed A-level Mathematics knowledge: Equation of a tangent
- 6993 · core · CA4: Equation of a tangent
- 4PM1 · core · 9F: Equation of a tangent
- 8365 · core · 4.4: Equation of a normal
- 9FM0 · core · Assumed A-level Mathematics knowledge: Equation of a normal
- 7367 · core · Assumed A-level Mathematics knowledge: Equation of a normal
- H245 · core · Assumed A-level Mathematics knowledge: Equation of a normal
- H645 · core · Assumed A-level Mathematics knowledge: Equation of a normal
- 6993 · core · CA4: Equation of a normal
- 4PM1 · core · 9F: Equation of a normal
Next practice: Stationary points.