Editing Euler's formula (section)

===Using differentiation===
This proof shows that the quotient of the trigonometric and exponential expressions is the constant function one, so they must be equal (the exponential function is never zero,<ref name=Apostol>{{cite book |last=Apostol |first=Tom |title=Mathematical Analysis |page=20 |publisher=Pearson |date=1974 |isbn=978-0201002881}} Theorem 1.42</ref> so this is permitted).<ref>user02138 (https://math.stackexchange.com/users/2720/user02138), How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?, URL (version: 2018-06-25): https://math.stackexchange.com/q/8612</ref>

Consider the function {{math|''f''(''θ'')}} 
<math display="block"> f(\theta) = \frac{\cos\theta + i\sin\theta}{e^{i\theta}} = e^{-i\theta} \left(\cos\theta + i \sin\theta\right) </math>
for real {{mvar|θ}}. Differentiating gives by the [[product rule]]
<math display="block"> f'(\theta) = e^{-i\theta} \left(i\cos\theta - \sin\theta\right) - ie^{-i\theta} \left(\cos\theta + i\sin\theta\right) = 0</math>
Thus, {{math|''f''(''θ'')}} is a constant. Since {{math|1=''f''(0) = 1}}, then {{math|1=''f''(''θ'') = 1}} for all real {{mvar|θ}}, and thus 
<math display="block">e^{i\theta} = \cos\theta + i\sin\theta.</math>