Implicit and Parametric Tangents

Curves defined implicitly

If \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\mathrm{f}(x)}{\mathrm{g}(x)}\), then

• the tangent is parallel to the \(x\)-axis when \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=0\), so solve \(\mathrm{f}(x)=0\) while checking \(\mathrm{g}(x)\ne0\);
• the tangent is parallel to the \(y\)-axis when the gradient is unbounded, so solve \(\mathrm{g}(x)=0\) while checking the curve and numerator conditions.

Parametric curves

For \(x=\mathrm{f}(t)\) and \(y=\mathrm{g}(t)\),

\[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t}=\frac{\mathrm{g}'(t)}{\mathrm{f}'(t)},\qquad \frac{\mathrm{d}x}{\mathrm{d}t}\ne0.\]