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

Answer 1

Aurora Alvarado

#= 9/x^2 e^ (-3/x)#

#d/dx 3e^ (-3/x)#

a few thoughts first

#d/dx alpha f(x) = alpha d/dx f(x) #

and #d/dx e^{g(x)} = g'(x) e^{g(x))# by the chain rule

so here we can say that

#d/dx 3e^ (-3/x)# #= 3 d/dx e^ (-3/x)# #= 3 d/dx (- 3/x) e^ (-3/x)#

#= 3 d/dx (- 3x^{-1}) e^ (-3/x)#

#= 3 (-1) (- 3x^{-1-1}) e^ (-3/x)# by the power rule

#= 3 * 3x^{-2} e^ (-3/x)#

#= 9/x^2 e^ (-3/x)#

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Answer 2

Scarlett Allison

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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Answer from HIX Tutor

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.