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The derivative for the parametric equations ##x=f(t)## and ##y=g(t)## is given by

##\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##.

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.

##\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 parametrised**even 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.

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