Examples

Slope of the tangent line to a parametric curve

Find a general formula for the slope of the tangent line to a parametric curve $\gamma (t)=(x(t),y(t))$ at the point determined by $t_0$.
Solution: Consider $t$ as a function of $x$. Remember the formula $\frac{dy}{dx}=\frac{1}{\frac{df}{dy}}$ when $x=f(y)$ so $y=f^{-1}(x)$. (Think about implicit differentiation.) Then using the Chain Rule and this formula, we have:

$$\frac{dy}{dx}=\frac{dy}{dt}\cdot \frac{dt}{dx}=\frac{dy}{dt}\cdot \frac{1}{\frac{dx}{dt}}=\frac{y^{\prime }(t)}{x^{\prime }(t)},$$

so the answer is $m(t_0)=\frac{y^{\prime }(t_0)}{x^{\prime }(t_0)}$.

Tangent line to a cycloid

A cycloid is given by the parametric curve with formula $(x,y)=(r(t-\sin (t)),r(1-\cos (t))$. Find the tangent line to this cycloid at $t=\pi /3$.
Solution: Using the formula for $\frac{dy}{dx}$, obtain:

$$\frac{dy}{dx}=\frac{r(0+\sin (t))}{r(1-\cos (t))}=\frac{\sin (t)}{1-\cos (t)}.$$

Then we have:

$$m={\frac{dy}{dx}|}_{t=\pi /3}=\sqrt{3}.$$

The image point when $t=\pi /3$ is $(x,y)=(r(\frac{\pi }{3}-\frac{\sqrt{3}}{2}),\frac{r}{2})$. After solving for the $y$-intercept we obtain the final equation:

$$y=\sqrt{3}x+r\left(\frac{\pi }{\sqrt{3}}-2\right).$$