When \(x\) and \(y\) are linked by an equation, differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\).

Differentiate each term; use the product rule where needed.

Collect all terms containing \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) and solve for it.

Substitute the coordinates, then form the tangent or normal equation.

For \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{N(x,y)}{D(x,y)}\), a horizontal tangent requires \(N=0\) and \(D\ne0\). A zero denominator is only a candidate for a vertical tangent: check the point lies on the curve, \(N\ne0\), and local behaviour. If both vanish, the formula alone does not decide the tangent.

Applications of Differentiation

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Tangents to Implicit Curves

Tangents to Implicit Curves