The tangent to a curve at a point has the same gradient as the curve there, which is f′(a) at x = a. The normal is perpendicular to the tangent, so its gradient is −1/f′(a). With a gradient and the point (a, f(a)), the equation follows from y − y₁ = m(x − x₁).
Tangents and normals
- Tangent gradient at x = a: m = f′(a)
- Normal gradient at x = a: −1/f′(a)
- Point on the curve: (a, f(a))
- Equation: y − y₁ = m(x − x₁)
The point comes from the original curve; the gradient comes from the derivative.
What are a tangent and a normal?
A tangent just touches a curve at a point and has exactly the same gradient as the curve there. A normal passes through the same point at right angles to the tangent. Differentiation gives the gradient; the rest is straight-line work.
Finding the gradient of a tangent
Differentiate to get f′(x), then substitute the x-value of the point. Differentiate first and substitute second: putting the number in before differentiating loses the gradient function entirely.
Equation of a tangent
You need two ingredients: the gradient m = f′(a) from the derivative, and the point (a, f(a)) from the original curve. If you are given only the x-value, substitute it into the curve's equation to get the y-coordinate. Then use y − y₁ = m(x − x₁) and rearrange into the form asked for.
Equation of a normal
The normal is perpendicular to the tangent, and perpendicular gradients multiply to −1. So flip the tangent gradient and change its sign: a tangent gradient of 2 gives a normal gradient of −½, and −3 gives ⅓. Use the same point on the curve.
Where tangents and normals meet the axes
Once you have the line, set y = 0 to find where it crosses the x-axis, or x = 0 for the y-axis. Mixing these up is the usual slip. Questions often go on to ask for the area of a triangle formed by a tangent, a normal and an axis.
Finding the point where the tangent has a given gradient
Sometimes you are given the gradient and must find the point. Set f′(x) equal to that gradient and solve, then substitute into the curve to get y. A tangent parallel to the line y = mx + c has the same gradient m. If f′(x) is quadratic there may be two points, so do not stop at one.
Worked examples
Example 1. Find the gradient of the tangent to y = x³ − 6x at x = 2.
- f′(x) = 3x² − 6.
- At x = 2: 3(4) − 6 = 6.
Answer: The tangent gradient is 6.
Marks: M1 for f′(x), A1 for 6.
Example 2. Find the equation of the tangent to y = x² − 4x + 5 at x = 3.
- Point: f(3) = 9 − 12 + 5 = 2, so (3, 2).
- Gradient: f′(x) = 2x − 4, so f′(3) = 2.
- Tangent: y − 2 = 2(x − 3), so y = 2x − 4.
Answer: y = 2x − 4.
Marks: B1 for the point, M1 for the line, A1 for y = 2x − 4.
Example 3. Find the equation of the normal to y = x² + 1 at x = 2.
- Point: f(2) = 5, so (2, 5).
- Tangent gradient f′(2) = 4, so the normal gradient is −¼.
- Normal: y − 5 = −¼(x − 2), so y = −¼x + 11/2.
Answer: y = −¼x + 11/2, or x + 4y − 22 = 0.
Marks: B1 for −¼, M1 for the line, A1 for the equation.
Example 4. The tangent to a curve is y = 4x − 9. Find where it crosses each axis.
- x-axis: 0 = 4x − 9, so x = 9/4.
- y-axis: x = 0 gives y = −9.
Answer: (9/4, 0) and (0, −9).
Marks: M1 for setting y = 0 and x = 0, A1 for both points.
Example 5. Find the point on y = x² − 6x + 4 where the tangent has gradient 2.
- f′(x) = 2x − 6. Set 2x − 6 = 2, so x = 4.
- f(4) = 16 − 24 + 4 = −4.
Answer: (4, −4).
Marks: M1 for 2x − 6 = 2, A1 for the point.
Common mistakes
- Using the derivative for the y-coordinate. The point lies on the curve, so find y from f(x), not f′(x).
- Forgetting to change the sign for the normal. The normal gradient is −1/m, not 1/m.
- Substituting before differentiating. Find f′(x) first, then substitute the x-value.
- Confusing the axes. The x-axis is y = 0; the y-axis is x = 0.
- Stopping at one point. If f′(x) = m is quadratic, check for a second solution.
How tangents and normals are examined
Tangent and normal questions appear on the Pure 1 paper (WMA11) of Edexcel International A-Level and in first-year Pure for Edexcel A-Level (9MA0). They are usually the later parts of a longer question:
- find dy/dx and the gradient at a given point
- find the equation of the tangent or normal, often with integer coefficients
- find where the line meets an axis or another line
- find the area of a triangle formed by tangents, normals and axes
Work through the Pure 1 differentiation exam questions, which include multi-part tangent and normal questions with mark schemes. The differentiation worksheet and the differentiation revision engine give further practice.
This combines basic differentiation with straight line equations.
Frequently asked questions
How do you find the equation of a tangent?
Differentiate, substitute the x-value to get the gradient, find the y-value from the curve, then use y − y₁ = m(x − x₁).
What is the gradient of a normal?
−1 divided by the gradient of the tangent at that point.
What is the difference between a tangent and a normal?
A tangent touches the curve with the same gradient as the curve. A normal passes through the same point at right angles to the tangent.