The derivative for the parametric equations ##x=f(t)## and ##y=g(t)## is given by(adsbygoogle = window.adsbygoogle || []).push({});

##\frac{dy}{dx}=\frac{\Big(\frac{dy}{dt}\Big)}{\Big(\frac{dx}{dt}\Big)}##

The proof of the above formula requires that ##y## be a function of ##x##, as seen in http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx

But somehow the formula works even if ##y## is not a function of ##x##, for example, it works for the unit circle ##x^2+y^2=1##, whose parametric equations are ##x=\cos t## and ##y=\sin t##.

Does the formula always work as long as the relation between ##x## and ##y## can be parametrisedeven if ##y## is not a function of ##x##? If not, what is a counterexample? For what parametric equations does the formula fail?

Note: A function is a relation that passes the vertical-line test.

Edit: We exclude cases where the curve intersects itself because the derivative doesn't exist at the intersecting points.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Does derivative formula work for all parametric equations

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - Does derivative formula | Date |
---|---|

I Why does this concavity function not work for this polar fun | Jan 26, 2018 |

B Why does this derivation fail? | May 28, 2017 |

B How does the delta ε definition prove derivatives? | Aug 9, 2016 |

I How does one derived the error bound for approximations? | Jun 27, 2016 |

B What does the derivative of a function at a point describe? | Mar 31, 2016 |

**Physics Forums - The Fusion of Science and Community**