Question

Shouldn't the derivative of e^(-x) be -xe^(-x-1)? why is it -e^(-x)

Answer 1

Because you have to mulpiply by the derivative of -x which is -1 You can't use the power rul e like that unless it's a constant number :P .

Answer 2

You can only use the power rule if the exponent is a constant number NOT a function.

Answer 3

so e is the constant here right?

Answer 4

yes. e is 2.7 something something something ... i think

Answer 5

so you can never have x as an exponent to use the chain rule?

Answer 6

even say (x+1) as an exponent?

Answer 7

to use the power rule. Chain rule is not the power rule.

Answer 8

sorry sorry. i meant power rule

Answer 9

right. just like using the power rule for
\[y=10^x\] would be inappropriate

Answer 10

e^u = e^u * u'

Answer 11

where u is more complicated than just x

Answer 12

@perl so the derivative of e^(x+1) would be e^(x+1) * (1)?

Answer 13

Recall:\[\frac{d}{dx}e^x=e^x\]

First example:
\[\frac{d}{dx}e^{-x}=e^{-x}\frac{d}{dx}(-x) = -e^{-x}\]
Second example:
\[\frac{d}{dx}e^{x+1}=e^{x+1}\frac{d}{dx}(x+1) = e^{x+1}\]
You can use substitution, if that makes it easier to understand.
let u = x+1
\[\frac{d}{dx}e^{x+1}=\frac{d}{du}e^u \times \frac{d}{dx}(x+1)= e^{u} \times 1 = e^{x+1}\]

Answer 14

wait. let me process it

Answer 15

ok. i think i get it now. it's just a rule that would in my opinion be better to learn than understand anyway

Answer 16

and if it was e^(-x+1) would it be -e^(-x+1)?

Answer 17

@Callisto

Answer 18

Yes :)
How do you get that?

Answer 19

i just learnt it!
if the power is negative then the constants sign will also change, but the exponent will stay the same

Answer 20

right?

Answer 21

and i wont be able to use power rule even if it is e^(2x)

Answer 22

?

Answer 23

You can actually use chain rule to get that.
\[y=e^{-x+1}\]
Let u = -x+1
\[\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = \frac{d}{du}(e^u) \times \frac{d}{dx}(-x+1)=-e^u = -e^{-x+1}\]

Answer 24

The negative sign comes from the fact that \[\frac{d}{dx} -x = -1\]
if for instance the question was
\[ \frac{d}{dx} e^{-2x+1}\] the answer would then be
\[-2e^{-2x+1}\]
because -2 is the derivative of -2x+1
if for instance the question was
\[ \frac{d}{dx} e^{-5x+1}\] the answer would then be
\[-5e^{-2x+1}\]
because -5 is the derivative of -5x+1

Answer 25

You're actually using chain rule instead of power rule in doing these examples.
For e^(2x)
\[y = e^{2x}\]
Let u = 2x
\[\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = \frac{d}{du}(e^u) \times \frac{d}{dx}(2x)=...?\]

Answer 26

@freewilly922 for e^(-5x+1) shouldn't it be -5e^(-5x+1)? you have -5e^(-2x+1)

Answer 27

yes it should I was copying and pasting and messed up. Good catch. And sorry.

Answer 28

@Callisto the derivative of 2x is 2. so should it be 2e^(2x)?

Answer 29

Yes.

Answer 30

good work @freewilly922 very good explanation.

Answer 31

YIPPEEE!!

Answer 32

i wish i could give 2 best responses

Answer 33

freewilly needs it more i guess

Answer 34

@Callisto was a better response.

Answer 35

u both did great. thank you very much @Callisto and @freewilly922