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Real exponential function/Derivative/Fact/Proof
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Real exponential function/Derivative/Fact
Proof
Due to
fact
, we have
exp
′
(
x
)
=
(
∑
n
=
0
∞
x
n
n
!
)
′
=
∑
n
=
1
∞
(
x
n
n
!
)
′
=
∑
n
=
1
∞
n
n
!
x
n
−
1
=
∑
n
=
1
∞
1
(
n
−
1
)
!
x
n
−
1
=
∑
n
=
0
∞
x
n
n
!
=
exp
x
.
{\displaystyle {}{\begin{aligned}\exp \!'(x)&={\left(\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\right)}'\\&=\sum _{n=1}^{\infty }{\left({\frac {x^{n}}{n!}}\right)}'\\&=\sum _{n=1}^{\infty }{\frac {n}{n!}}x^{n-1}\\&=\sum _{n=1}^{\infty }{\frac {1}{(n-1)!}}x^{n-1}\\&=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\\&=\exp x.\end{aligned}}}
To fact