# How do you use the Quotient Rule to differentiate the function #f (t) = (cos (7t)) / t^5#?

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To differentiate the function ( f(t) = \frac{\cos(7t)}{t^5} ) using the Quotient Rule, follow these steps:

- Identify ( u(t) ) as the numerator, ( v(t) ) as the denominator.
- Apply the Quotient Rule formula: ( \frac{d}{dt} \left( \frac{u(t)}{v(t)} \right) = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} ).
- Find the derivatives of ( u(t) ) and ( v(t) ).
- Substitute the derivatives and original functions into the Quotient Rule formula.
- Simplify the expression.

Applying this to ( f(t) = \frac{\cos(7t)}{t^5} ):

( u(t) = \cos(7t) ) and ( v(t) = t^5 ).

( u'(t) = -7\sin(7t) ) and ( v'(t) = 5t^4 ).

Now, using the Quotient Rule formula:

( f'(t) = \frac{(-7\sin(7t))(t^5) - (\cos(7t))(5t^4)}{(t^5)^2} )

( f'(t) = \frac{-7t^5\sin(7t) - 5t^4\cos(7t)}{t^{10}} )

( f'(t) = \frac{-7t^5\sin(7t) - 5t^4\cos(7t)}{t^{10}} )

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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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