Define the (possibly complex) function ƒ("x"), of real variable "x", as

The derivative of ƒ("x"), according to the product rule, is:

Therefore, ƒ("x") must be a constant function in "x". Because ƒ(0) is known, the constant that ƒ("x") equals for all real "x" is also known. Thus, 

Multiplying both sides by "e" and using

it follows that

Q.E.D.

Using calculus
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Differential equation definition
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