# What is the derivative of #f(t) = (t/sint , cost/t^2 ) #?

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To find the derivative of the function ( f(t) = \left( \frac{t}{\sin t}, \frac{\cos t}{t^2} \right) ), we differentiate each component separately using the quotient rule and the chain rule.

The derivative of the first component with respect to ( t ) is: [ \frac{d}{dt} \left( \frac{t}{\sin t} \right) = \frac{\sin t - t\cos t}{\sin^2 t} ]

The derivative of the second component with respect to ( t ) is: [ \frac{d}{dt} \left( \frac{\cos t}{t^2} \right) = \frac{-2\cos t - t(-\sin t)}{t^4} = \frac{2\cos t + t\sin t}{t^4} ]

Therefore, the derivative of ( f(t) ) is ( \left( \frac{\sin t - t\cos t}{\sin^2 t}, \frac{2\cos t + t\sin t}{t^4} \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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