d/dx (e^x)
e^x
The derivative of e^x with respect to x is e^x.
We can use the definition of the derivative to prove this statement:
f(x) = e^x
f'(x) = limit as h approaches 0 of (f(x+h) – f(x))/h
= limit as h approaches 0 of ((e^(x+h) – e^x)/h)
= limit as h approaches 0 of ((e^x * e^h – e^x)/h)
= limit as h approaches 0 of (e^x * (e^h – 1)/h)
We note that the limit of (e^h – 1)/h as h approaches 0 is equal to 1. Therefore,
= limit as h approaches 0 of e^x * 1
= e^x
Therefore, the derivative of e^x with respect to x is e^x.
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