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Question Number 211944 by liuxinnan last updated on 25/Sep/24

$$$$$$f\left(x\right)=\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}{f}^{{}^{\prime }}\left(x\right)=?$$$$$$$$$$

Answered by efronzo1 last updated on 25/Sep/24

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{{(\sqrt{1+\mathrm{x}}-\sqrt{1-\mathrm{x}})}^2}{2\mathrm{x}}$$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{2-2\sqrt{1-{\mathrm{x}}^2}}{2\mathrm{x}}=\frac{1}{\mathrm{x}}-\sqrt{\frac{1}{{\mathrm{x}}^2}-1}$$$$\mathrm{f}{}^{\prime }\left(\mathrm{x}\right)=-\frac{1}{{\mathrm{x}}^2}-\frac{-\frac{2}{{\mathrm{x}}^3}}{2\sqrt{\frac{1}{{\mathrm{x}}^2}-1}}$$

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