Home > Calculus > Differentiating Exponential Functions with Base e

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

Answer 1

Evelyn Agard

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

using the chain rule

#(dy)/(dx)=(dy)/(du)color(red)((du)/(dx))#

#y=e^(-3x)#

#color(red)(u=-3x=>(dy)/(du)=-3)#

#(dy)/(du)=d/(du)(e^u)=e^u#

#:.(dy)/(dx)=(dy)/(du)color(red)((du)/(dx))=e^uxxcolor(red)((-3))#

#=-3e^u=-3e^(-3x)#

in general:

#d/(dx)(e^(f(x)))=f'(x)e^(f(x))#

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Answer 2

Ezra Adams

#color(brown)(f'(x) = -3 e^-(3x)#

#f(x) = e^-(3x)#

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

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

#f'(x) = e^-(3x) * -3#

#color(brown)(f'(x) = -3 e^-(3x)#

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Answer 3

Cameron Alexander

To find the derivative of ( e^{-3x} ), you can use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

The derivative of ( e^{-3x} ) with respect to ( x ) is ( -3e^{-3x} ).

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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.