Parametric Derivatives — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find dy/dx for the parametric curve.

   $$x=t,y=t^2$$

   Answer: $2t$

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{2t}{1}=2t$$
2. 2.

   Find dy/dx for the parametric curve.

   $$x=t^2,y=t^3$$

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

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{3t^2}{2t}=\frac{3t}{2}$$
3. 3.

   Find dy/dx for the parametric curve.

   $$x=\cos \left(t\right),y=\sin \left(t\right)$$

   Answer: $-\frac{1}{\tan \left(t\right)}$

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{\cos \left(t\right)}{-\sin \left(t\right)}=-\frac{1}{\tan \left(t\right)}$$
4. 4.

   Find dy/dx for the parametric curve.

   $$x=t+1,y=t^3-t$$

   Answer: $3t^2-1$

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{3t^2-1}{1}=3t^2-1$$
5. 5.

   Find dy/dx for the parametric curve.

   $$x=e^t,y=e^{-t}$$

   Answer: $-e^{-2t}$

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{-e^{-t}}{e^t}=-e^{-2t}$$
6. 6.

   Find dy/dx for the parametric curve.

   $$x=t^2+1,y=t^2-t$$

   Answer: $\frac{t-\frac{1}{2}}{t}$

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{2t-1}{2t}=\frac{t-\frac{1}{2}}{t}$$
7. 7.

   Find dy/dx for the parametric curve.

   $$x=t,y=t^3$$

   Answer: $3t^2$

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{3t^2}{1}=3t^2$$
8. 8.

   Find dy/dx for the parametric curve.

   $$x=2t,y=t^2$$

   Answer: $t$

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{2t}{2}=t$$
9. 9.

   Find dy/dx for the parametric curve.

   $$x=t^2,y=t^4$$

   Answer: $2t^2$

   Step-by-step solution

   1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
   2. $$\frac{4t^3}{2t}=2t^2$$
10. 10.

    Find dy/dx for the parametric curve.

    $$x=\sin \left(t\right),y=\cos \left(t\right)$$

    Answer: $-\tan \left(t\right)$

    Step-by-step solution

    1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
    2. $$\frac{-\sin \left(t\right)}{\cos \left(t\right)}=-\tan \left(t\right)$$
11. 11.

    Find dy/dx for the parametric curve.

    $$x=t-1,y=t^2+t$$

    Answer: $2t+1$

    Step-by-step solution

    1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
    2. $$\frac{2t+1}{1}=2t+1$$
12. 12.

    Find dy/dx for the parametric curve.

    $$x=3t,y=t^3$$

    Answer: $t^2$

    Step-by-step solution

    1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
    2. $$\frac{3t^2}{3}=t^2$$
13. 13.

    Find dy/dx for the parametric curve.

    $$x=e^t,y=t^2$$

    Answer: $2t{e}^{-t}$

    Step-by-step solution

    1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
    2. $$\frac{2t}{e^t}=2t{e}^{-t}$$
14. 14.

    Find dy/dx for the parametric curve.

    $$x=t^3,y=t^2$$

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

    Step-by-step solution

    1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
    2. $$\frac{2t}{3t^2}=\frac{2}{3t}$$
15. 15.

    Find dy/dx for the parametric curve.

    $$x=t+2,y=t^3+1$$

    Answer: $3t^2$

    Step-by-step solution

    1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
    2. $$\frac{3t^2}{1}=3t^2$$
16. 16.

    Find dy/dx for the parametric curve.

    $$x=\cos \left(2t\right),y=\sin \left(2t\right)$$

    Answer: $-\frac{1}{\tan \left(2t\right)}$

    Step-by-step solution

    1. $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$
    2. $$\frac{2\cos \left(2t\right)}{-2\sin \left(2t\right)}=-\frac{1}{\tan \left(2t\right)}$$

More from Calculus II

• Integration by Parts
• Trigonometric Integrals
• Partial Fractions
• Improper Integrals
• Sequence Limits
• Geometric Series
• p-Series
• Convergence Tests
• Power Series
• Taylor Series
• Parametric Arc Length
• Polar Area