Difference between revisions of "Derivative of the exponential function"
From specialfunctionswiki
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\dfrac{\mathrm{d}}{\mathrm{d}z} e^z = e^z,$$ | $$\dfrac{\mathrm{d}}{\mathrm{d}z} e^z = e^z,$$ | ||
where $e^z$ denotes the [[exponential function]]. | where $e^z$ denotes the [[exponential function]]. | ||
| − | + | ||
| − | + | ==Proof== | |
$$e^z = \displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{k!}.$$ | $$e^z = \displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{k!}.$$ | ||
[[Term-by-term differentiation]] of this sum shows | [[Term-by-term differentiation]] of this sum shows | ||
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\end{array}$$ | \end{array}$$ | ||
as was to be shown. █ | as was to be shown. █ | ||
| − | + | ||
| − | + | ==References== | |
Revision as of 03:43, 6 June 2016
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} e^z = e^z,$$ where $e^z$ denotes the exponential function.
Proof
$$e^z = \displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{k!}.$$ Term-by-term differentiation of this sum shows $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} e^z &= \displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{d}}{\mathrm{d}z} \left[ \dfrac{z^k}{k!} \right] \\ &=\displaystyle\sum_{k=1}^{\infty} \dfrac{z^{k-1}}{(k-1)!} \\ &=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{k!} \\ &=e^z, \end{array}$$ as was to be shown. █
