# How do you differentiate #f(x) = (cos2x)/(e^(2x)-x)# using the quotient rule?

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To differentiate ( f(x) = \frac{\cos^2(2x)}{e^{2x} - x} ) using the quotient rule, you would apply the formula:

[ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ]

Where ( u(x) = \cos^2(2x) ) and ( v(x) = e^{2x} - x ). Then, differentiate each function separately:

[ u'(x) = -4\sin(2x)\cos(2x) ] [ v'(x) = 2e^{2x} - 1 ]

Finally, plug these into the quotient rule formula to get the derivative of ( f(x) ):

[ f'(x) = \frac{-4\sin(2x)\cos(2x)(e^{2x} - x) - \cos^2(2x)(2e^{2x} - 1)}{(e^{2x} - x)^2} ]

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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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