I'm trying to understand the proof to Euler's formula and I keep getting stuck on one of the steps.
We start with:
$${\displaystyle e^{ix}=r(\cos \theta +i\sin \theta ).}$$
We will then assume that this will hold true for some unknown values for $\theta$ and $r$, and try to reveal those values.
In the next step we take the derivative of both sides, so that:
$${\displaystyle ie^{ix}=(\cos \theta +i\sin \theta ){\frac {dr}{dx}}+r(-\sin \theta +i\cos \theta ){\frac {d\theta }{dx}}.}$$
....and this is where they lose me. $r(\cos \theta +i\sin \theta )$ is (as far as I can see) "not" a product of two functions, so why on earth are they using the product rule instead of the chain rule?
Link:
https://en.wikipedia.org/wiki/Euler%27s_formula#Using_power_series
$r(\cos(\theta)+i\sin(\theta)$ is indeed the product of two functions of $x$, $$r(x)$$ and $$g(x)=(\cos(\theta(x))+i\sin(\theta(x)))$$ So the product rule on these functions is $$r'(x)g(x)+r(x)g'(x)$$ which gives what you have above, $$ \frac{dr}{dx}(\cos(\theta(x))+i\sin(\theta(x))) + r(-\sin(\theta(x))+i\cos(\theta(x)))\frac{d\theta}{dx} $$
Just to show the product rule with your example in the comments, given $$2(x+5)$$ the derivative using product rule is $$0\cdot(x+5)+2\cdot(1)=2$$