# How do you differentiate # x/e^(3x)#?

Apply the quotient rule to get

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To differentiate ( \frac{x}{e^{3x}} ), you can use the quotient rule. The quotient rule states that if you have a function in the form ( \frac{f(x)}{g(x)} ), then its derivative is given by:

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

For ( \frac{x}{e^{3x}} ), let ( f(x) = x ) and ( g(x) = e^{3x} ). Then:

[ f'(x) = 1 ] [ g'(x) = 3e^{3x} ]

Applying the quotient rule:

[ \frac{d}{dx} \left( \frac{x}{e^{3x}} \right) = \frac{(1)(e^{3x}) - (x)(3e^{3x})}{(e^{3x})^2} ]

[ = \frac{e^{3x} - 3xe^{3x}}{e^{6x}} ]

[ = \frac{e^{3x}(1 - 3x)}{e^{6x}} ]

[ = \frac{1 - 3x}{e^{3x}} ]

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