Calculus

Tangents and Normals: Equations

Find tangent and normal lines to a curve using derivative gradients.

Core method

For \(y=f(x)\), the tangent gradient at \(x=a\) is \(f'(a)\). The normal is perpendicular to the tangent.

Tangent gradient\[m_T=f'(a)\]
Normal gradient\[m_N=-\frac{1}{m_T}\]

Then use \(y-y_1=m(x-x_1)\).

Worked example

Find the tangent and normal to \(y=x^2-1\) at \(P(2,3)\).

  1. \(\frac{dy}{dx}=2x\).
  2. At \(x=2\), \(m_T=4\).
  3. Tangent: \(y-3=4(x-2)\Rightarrow y=4x-5\).
  4. Normal gradient: \(m_N=-\frac14\).
  5. Normal: \(y-3=-\frac14(x-2)\Rightarrow y=-\frac14x+\frac72\).
Exam tip: if the tangent is horizontal, the normal is vertical.

Tangents and normals

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Equation of a tangent

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Equation of a normal

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