e^x derivative using first principles

Question

Using the Taylor series expansion for e^x , show that the derivative of e^x (using first principles) is ex (include at least three terms of the series to confirm the derivative of e^x)

Answer 1

Note that

$$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+...$$

thus

$$(e^x)’=0+1+2\frac{x}{2}+3\frac{x^3}{3!}+...=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+...=e^x$$