- #1

LearninDaMath

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## Homework Statement

I am of the conclusion that, under any circumstance, the extended exponential rule can not be applied to (1+x)[itex]^{1/x}[/itex].Thus, there is no way for the extended exponential rule to arise when taking the derivative of:

f(x) = (1+x)[itex]^{\frac{1}{x}}e^{x}[/itex]

For instance, if my first step for finding the derivative of this function was to apply the product rule, i'd get:

f'x = ((1+x)[itex]^{\frac{1}{x}})'(e^{x})[/itex] + ((1+x)[itex]^{\frac{1}{x}})(e^{x})'[/itex]

And in the next step, if I were to take the derivatives by first applying the exponential rule to (1+x)[itex]^{1/x}[/itex],

I would get an incorrect outcome because while I could apply the exponential rule to something like b^x, I would not be able to apply exponential rule to something like (b+x)^x

Is this correct so far?