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Question Number 26288 by sakshiraj@gmail.com last updated on 23/Dec/17

$$\frac{\mathit{d}}{\mathit{d}\mathit{x}}\frac{\mathit{x}-4}{2\sqrt{\mathit{x}}}$$

Answered by Joel578 last updated on 24/Dec/17

$$\mathrm{Use}\mathrm{quotient}\mathrm{rule}$$$$u=x-4\to u^{\prime }=1$$$$v=2\sqrt{x}\to v^{\prime }=\frac{1}{\sqrt{x}}$$$$$$$$\frac{d}{dx}\left(\frac{x-4}{2\sqrt{x}}\right)=\frac{u^{\prime }.v-v^{\prime }.u}{v^2}$$$$=\frac{2\sqrt{x}-\frac{1}{\sqrt{x}}(x-4)}{4x}$$$$=\frac{2\sqrt{x}-\frac{x}{\sqrt{x}}+\frac{4}{\sqrt{x}}}{4x}$$$$=\frac{x+4}{4x\sqrt{x}}=\frac{(x+4)\sqrt{x}}{4x^2}$$

Answered by Rasheed.Sindhi last updated on 27/Dec/17

$$\mathrm{An}\mathrm{Other}\mathrm{Way}$$$$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{x}-4}{2\sqrt{\mathrm{x}}}\right)$$$$=\frac{\mathrm{d}}{\mathrm{dx}}(\frac{\mathrm{x}}{{2\mathrm{x}}^{1/2}}-\frac{4}{{2\mathrm{x}}^{1/2}})$$$$=\frac{\mathrm{d}}{\mathrm{dx}}(\frac{1}{2}{\mathrm{x}}^{1/2}-{2\mathrm{x}}^{-1/2})$$$$=\frac{1}{2}\left(\frac{1}{2}{\mathrm{x}}^{1/2-1}\right)-2(-\frac{1}{2}{\mathrm{x}}^{-1/2-1})$$$$=\frac{1}{4}{\mathrm{x}}^{-1/2}+{\mathrm{x}}^{-3/2}$$$$=\frac{1}{{4\mathrm{x}}^{1/2}}+\frac{1}{{\mathrm{x}}^{3/2}}$$$$=\frac{1}{4\sqrt{\mathrm{x}}}+\frac{1}{\mathrm{x}\sqrt{\mathrm{x}}}$$$$=\frac{4+\mathrm{x}}{4\mathrm{x}\sqrt{\mathrm{x}}}\times \frac{\sqrt{\mathrm{x}}}{\sqrt{\mathrm{x}}}=\frac{(\mathrm{x}+4)\sqrt{\mathrm{x}}}{{4\mathrm{x}}^2}$$

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