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How do you differentiate the following parametric equation: # x(t)=t^2-t, y(t)=e^t+cost #?

Answer 1

Charles Arocha

#dy/dx=(e^t-sint)/(2t-1)#

For #color(blue)"parametric equations"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(dy/dx=(y'(t))/(x'(t)))color(white)(2/2)|)))#

#x(t)=t^2-trArrx'(t)=2t-1#

#y(t)=e^t+costrArry'(t)=e^t-sint#

#rArrdy/dx=(e^t-sint)/(2t-1)#

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Answer 2

Charles Arocha

To differentiate the parametric equations ( x(t) = t^2 - t ) and ( y(t) = e^t + \cos(t) ) with respect to ( t ), compute the derivatives of ( x ) and ( y ) separately using the chain rule if necessary. This yields ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ), which represent the rates of change of ( x ) and ( y ) with respect to ( t ), respectively.

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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.