# How do you differentiate the following parametric equation: # x(t)=(t-1)^2-e^t, y(t)= (t+2)^2+t^2#?

To find derivatives with respect to t, differentiate normally. To find

And

Rearranging, we can get...

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To differentiate the given parametric equations (x(t) = (t - 1)^2 - e^t) and (y(t) = (t + 2)^2 + t^2), follow these steps:

- Differentiate each parametric equation separately with respect to (t).
- Apply the chain rule where necessary.
- Express (dx/dt) and (dy/dt) as functions of (t).

Here are the derivatives:

For (x(t) = (t - 1)^2 - e^t):

[ \frac{dx}{dt} = 2(t - 1) - e^t ]

For (y(t) = (t + 2)^2 + t^2):

[ \frac{dy}{dt} = 2(t + 2) + 2t ]

These are the derivatives of the given parametric equations with respect to (t).

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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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