# What is the derivative of #f(t) = (tlnt, 3t^2-t ) #?

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To find the derivative of ( f(t) = (t \ln t, 3t^2 - t) ), we'll compute the derivative of each component function separately using the chain rule for the first function ( t \ln t ) and the power rule for the second function ( 3t^2 - t ).

[ \frac{d}{dt}(t \ln t) = t \frac{d}{dt}(\ln t) + \ln t \frac{d}{dt}(t) ]

[ = t \left( \frac{1}{t} \right) + \ln t (1) ]

[ = 1 + \ln t ]

[ \frac{d}{dt}(3t^2 - t) = 6t - 1 ]

So, the derivative of ( f(t) = (t \ln t, 3t^2 - t) ) with respect to ( t ) is:

[ f'(t) = \left( 1 + \ln t, 6t - 1 \right) ]

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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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