Difference between revisions of "Derivative of inverse error function"
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(Created page with "==Theorem== The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{erf}^{-1}(x)=\dfrac{\sqrt{\pi}}{2}e^{[\mathrm{erf}^{-1}(x)]^2},$$ where $\mathrm{erf}^{-1}$...") |
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{erf}^{-1}(x)=\dfrac{\sqrt{\pi}}{2} | + | $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{erf}^{-1}(x)=\dfrac{\sqrt{\pi}}{2}\exp \left({[\mathrm{erf}^{-1}(x)]^2} \right),$$ |
| − | where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]] and $\ | + | where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]], $\pi$ denotes [[pi]], and $\exp$ denotes the [[exponential]]. |
==Proof== | ==Proof== | ||
Latest revision as of 04:38, 16 September 2016
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{erf}^{-1}(x)=\dfrac{\sqrt{\pi}}{2}\exp \left({[\mathrm{erf}^{-1}(x)]^2} \right),$$ where $\mathrm{erf}^{-1}$ denotes the inverse error function, $\pi$ denotes pi, and $\exp$ denotes the exponential.
