Taylor series approximations work because the derivatives of the taylor series expansion and the function it is approximating are the same (and the values at the given point). Likewise, the derivative of e^x is e^x because the taylor expansion of e^x is 1 + x/1! + x^2/2! + x^3/3!.. which becomes 0 + 1 + x/1! + x^2/2! + x^3/3!.. when the derivative is taken. So the derivative of the taylor expansion of e^x is itself.