Build a tangent
Find the tangent to y = x² + 1 at x = 2.
- Differentiate: dy/dx = 2x, giving gradient 4.
- The curve gives the point (2, 5).
- Use y − 5 = 4(x − 2).
Answer: y = 4x − 3
Tangents and normals
Use the derivative for a gradient and the original curve for a point.
Before you start
You’ll need Differentiating powers, Straight lines and gradients. Follow a link if you want to revise it first.
Finding the point and gradient
The derivative is a function giving the gradient at each input. Its value at a specific x is a number.
Substitute into the original curve to find the y coordinate. Substituting into the derivative gives a gradient instead.
A normal is perpendicular to the tangent. Its gradient is −1/m when the tangent gradient m is nonzero; a horizontal tangent has a vertical normal.
Both lines below pass through (2, 5). Their gradients multiply to −1.
The tangent and normal at (2, 5)
Build a normal
Find the normal at the same point.
- The normal gradient is −¼.
- Use the same point (2, 5): y − 5 = −¼(x − 2).
Answer: y = −x/4 + 11/2
The derivative supplies the gradient, not the y coordinate of the point.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
Practice is filtered to your selected course. Change course or options when needed.
Check your understanding
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How would you write the normal when the tangent gradient is zero?