# Complex derivatives

I have to solve an ODE with variation of coefficient technique. It's pretty easy but I have no clue what is the first and second derivative of e^ix and e^-ix.

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arildno
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If i had been a real number, what would the first and second derivatives have been then?

e^ix
first
i*e^ix
second
i^2*e^ix

e^-ix
first
-i*e^-ix
second
i^2*e^-ix

p.s. I've read about the Cauchy-Riemann equation but just not sure how to apply it... should I split the exponential in a sin and a cos?
p.s.s. There are probably rules, like exponential function are always derivable or something but I'm not fallowing any complex variables class right now so any insight is appreciated...

Last edited:
arildno
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e^ix
first
i*e^ix
second
i^2*e^ix

e^-ix
first
-i*e^-ix
second
i^2*e^-ix

EXACTLY!
And that is precisely what holds when "i" is a complex/imaginary number as well! • CivilSigma
Gib Z
Homework Helper
When dealing with these things, forget i is anything, just remember its a constant. Then after the actual differentiation, you can remember what it is.

Yeah. If

$$\exp(ix),\,\,\,x\in \mathbb{R},$$

(which is what it looks like you have) then it's what the above two said. But if you have

$$\exp(iz),\,\,\,z\in \mathbb{Z},$$

you need to be more careful. Let us know if that is indeed what you have.

mathwonk
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2020 Award
what are you doing in a de course ifm you do not know the derivative of e^z?

I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.

HallsofIvy
Homework Helper
I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.
That is exactly what has been answered in each of these responses. For any constant, a, the derivative of $e^{ax}$ is $ae^{ax}$.

That is a result of the very basic fact that the derivative of $e^x$ is $e^x$ (world's easiest derivative!) and the chain rule.

I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.

$$\frac{d}{dx}(e^{jx})=je^{jx}$$
$$\frac{d^2}{dx^2}(e^{jx})=-e^{jx}$$

mathwonk
Actually I myself was once in an ode course when I had forgot the derivative of e^x. My solution was to go get a Schaum's outline series of ode and do a lot of problems and review my \$ off.