# How do you find the derivative of #3e^ (-3/x)#?

a few thoughts first

so here we can say that

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To find the derivative of (3e^{-3/x}), you can use the chain rule. The derivative is:

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

Using the chain rule and derivative of (-3/x):

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

Substituting this back into the original expression:

[ 3 \cdot e^{-3/x} \cdot \frac{3}{x^2} = \frac{9}{x^2} \cdot e^{-3/x} ]

So, the derivative of (3e^{-3/x}) is (\frac{9}{x^2} \cdot e^{-3/x}).

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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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