Question

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

Answer 1

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

Explanation:

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))`

Answer 2

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

Explanation:

`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)`