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3.14 Derivative of the exponential function

Example

The derivative of the exponential function is ${{d}\over{dx}}e^{x}=e^{x}$ .

The key observation is that

{{d}\over{dx}}{{x^{n}}\over{n!}}={{nx^{n-1}}\over{n!}}={{x^{n-1}}\over{(n-1)!}};

so when we differentiate the exponential series term-by-term, we have

{{d}\over{dx}}e^{x}={{d}\over{dx}}\Bigl(1+x+{{x^{2}}\over{2!}}+{{x^{3}}\over{3% !}}+\dots+{{x^{n}}\over{n!}}+\dots\Bigr)

{}\qquad=0+{{dx}\over{dx}}+{{d}\over{dx}}{{x^{2}}\over{2!}}+{{d}\over{dx}}{{x^% {3}}\over{3!}}+\dots+{{d}\over{dx}}{{x^{n}}\over{n!}}+\dots

{}\qquad=1+x+{{x^{2}}\over{2!}}+\dots+{{x^{n-1}}\over{(n-1)!}}+\dots=e^{x}.