# How do you differentiate the following parametric equation: # x(t)=t/(t-4), y(t)=1+t #?

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To differentiate the parametric equations ( x(t) = \frac{t}{t-4} ) and ( y(t) = 1 + t ) with respect to ( t ):

- Differentiate ( x(t) ) with respect to ( t ):

Using the quotient rule: [ \frac{d}{dt}\left(\frac{u(t)}{v(t)}\right) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2} ]

Let ( u(t) = t ) and ( v(t) = t-4 ).

[ x'(t) = \frac{(1)(t-4) - (t)(1)}{(t-4)^2} ] [ x'(t) = \frac{-4}{(t-4)^2} ]

- Differentiate ( y(t) ) with respect to ( t ):

[ y'(t) = \frac{d}{dt}(1 + t) ] [ y'(t) = 1 ]

Thus, the derivatives are: [ x'(t) = \frac{-4}{(t-4)^2} ] [ y'(t) = 1 ]

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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.

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.

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.

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