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How do you differentiate the following parametric equation: # x(t)=e^t-cost-t, y(t)=e^t-tsint #?

Answer 1

Elijah Abram

Derivative is #(e^t-sint-tcost)/(e^t+sint-1)#

The differential of a parametric equation of the type #f(x(t),y(t))# is given by #((dy)/(dt))/((dx)/(dt))#.

Here #x(t)=e^t-cost-t# hence #(dx)/(dt)=e^t+sint-1#

and as #y(t)=e^t-tsint#, #(dy)/(dt)=e^t-sint-tcost#

Hence #(df)/(dx)=(e^t-sint-tcost)/(e^t+sint-1)#

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Answer 2

Luke Alvarez

To differentiate the given parametric equations (x(t) = e^t - \cos(t) - t) and (y(t) = e^t - t \sin(t)) with respect to (t), you can use the chain rule and the product rule. The derivatives are:

[ \frac{dx}{dt} = e^t + \sin(t) - 1 ]

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

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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.