Differentiating parametric equations

Differentiate $x$ and $y$ separately with respect to $t$, then divide $\frac{dy}{dt}$ by $\frac{dx}{dt}$. The answer can stay in terms of $t$.

It is easy to flip the fraction, so check that the derivative of $y$ is on top and the derivative of $x$ is on the bottom. The formula does not apply at values of $t$ where $\frac{dx}{dt}=0$.

Examples

1. 01What is dy/dx?$\displaystyle x=t^3,\, y=t^2$AnswerHide

   Answer$\displaystyle \frac{2}{3t}$

   Steps

   1. Make the two derivatives with respect to $t$
   2. $\displaystyle \frac{dx}{dt}$$\displaystyle = 3t^{2}$$\displaystyle \frac{dy}{dt}$$\displaystyle = 2t$
   3. Divide, with the $y$ one on top
   4. $\displaystyle \frac{dy}{dx}$$\displaystyle = \frac{2t}{3t^{2}} = \frac{2}{3t}$
2. 02What is dy/dx?$\displaystyle x=t^2,\, y=t^3$AnswerHide

   Answer$\displaystyle \frac{3}{2}t$

   Steps

   1. Make the two derivatives with respect to $t$
   2. $\displaystyle \frac{dx}{dt}$$\displaystyle = 2t$$\displaystyle \frac{dy}{dt}$$\displaystyle = 3t^{2}$
   3. Divide, with the $y$ one on top
   4. $\displaystyle \frac{dy}{dx}$$\displaystyle = \frac{3t^{2}}{2t} = \frac{3}{2}t$
3. 03What is dy/dx?$\displaystyle x=t^{2}+1,\, y=t^{3}-3t$AnswerHide

   Answer$\displaystyle \frac{3t^{2}-3}{2t}$
4. 04What is dy/dx?$\displaystyle x=\cos t,\, y=\sin t$AnswerHide

   Answer$\displaystyle -\frac{\cos t}{\sin t}$
5. 05What is dy/dx?$\displaystyle x=2t-1,\, y=t^{2}+4t$AnswerHide

   Answer$\displaystyle t+2$

FAQ

Q1Do I need to rewrite the answer in terms of $x$?AnswerHide

A

Usually not; the answer is given in terms of $t$. For the slope at a particular point, substitute the value of $t$ for that point.

Q2I keep writing $\frac{dx/dt}{dy/dt}$ instead.AnswerHide

A

Think of it as $\frac{dy}{dx}=\frac{dy}{dt}\div\frac{dx}{dt}$: the $dt$ cancels, which confirms which derivative goes on top.