Grade 12 · Mathematics

Find the equation of a tangent

A tangent touches a differentiable curve at a point and has the same gradient as the curve there. To find its equation in Grade 12 Mathematics, calculate the point on the original function and the gradient from its derivative. These are different calculations. Then substitute both into the straight-line form y − y₁ = m(x − x₁).

Original explanation prepared with AI assistance.

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What you will practise

Combine a point and derivative gradient to find a tangent equation.

The method

Use f(a) for the y-coordinate and f′(a) for the gradient.

Worked example

Find the tangent to y = x² at x = 3.

The point is (3; 9) because 3² = 9.
Differentiate: dy/dx = 2x. At x = 3, m = 6.
Use y − 9 = 6(x − 3).
Expand to obtain y = 6x − 9. Check that (3; 9) satisfies the line.

Common mistake

Do not use f′(3) as the point’s y-coordinate. The derivative supplies the gradient.

Try it yourself

Find the tangent to y = x² + 1 at x = 2.

Show the explained solution

The point is (2; 5), m = 2(2) = 4, and y − 5 = 4(x − 2), so y = 4x − 3.

Curriculum and sources

Use the official CAPS subject document and your teacher’s annual plan to confirm the sequence and scope for your class. The link below is an official reference, not evidence of DBE approval of this explanation.

DBE: CAPS for the FET phase

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