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How do you differentiate the following parametric equation: # x(t)=cos^2t, y(t)=sint/t #?

Answer 1

Isaiah Allen

#dy/dx=-1/(2tsint)+1/(2t^2cost)#

Given that the equations are #x(t)=cos^2t#, #y(t)=sint/t#

The derivative of a parametric equation when in the form #dy/dx# can be re-written as #(dy//dt)/(dx//dt)#

For #y(t)# using quotient rule, #dy/dt=(tcost-sint)/t^2#

Similarly, for #x(t)#, using product rule, #dx/dt=-2costsint#

So, the parametric differentiative turns to #{(tcost-sint)//t^2}/{-2sintcost}#

Simplifying it shouldn't be a hassle to simplify.

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Answer 2

Ezra Adams

To differentiate the parametric equations (x(t) = \cos^2(t)) and (y(t) = \frac{\sin(t)}{t}), you need to find the derivatives of (x) and (y) with respect to (t).

[ \frac{dx}{dt} = -2\cos(t)\sin(t) ] [ \frac{dy}{dt} = \frac{t\cos(t) - \sin(t)}{t^2} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.