# How do you differentiate # x/(e^(2x))#?

Quotient rule :

Chain rule :

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To differentiate the function ( \frac{x}{e^{2x}} ), you can use the quotient rule.

The quotient rule states that if you have a function of the form ( \frac{u}{v} ), where ( u ) and ( v ) are differentiable functions of ( x ), then the derivative is given by: [ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} ]

Using this rule, differentiate ( \frac{x}{e^{2x}} ):

[ u = x ] [ v = e^{2x} ]

[ u' = 1 ] [ v' = 2e^{2x} ]

So applying the quotient rule:

[ \frac{d}{dx} \left( \frac{x}{e^{2x}} \right) = \frac{(1)(e^{2x}) - (x)(2e^{2x})}{(e^{2x})^2} ]

[ = \frac{e^{2x} - 2xe^{2x}}{e^{4x}} ]

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

[ = \frac{1 - 2x}{e^{2x}} ]

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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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