Yes, this is true. See the explanation below...
This answer many not be as formal as a proper mathematical proof, but I hope it will do.
According to the chain rule, when we differentiate
When we differentiate a second time. the process repeats itself, as the differentiation of the exponential yields another factor of 2:
(In truth, this is an application of the product rule, but I have omitted the second term, as the derivative of 2 is zero.)
This pattern continues for as many times as you choose to do the differentiation. Each derivative places another factor of 2 in front of the result, so that after
Use an inductive proof.
Then differentiate again:
So if it is true for
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