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\( x\left(x^{2}+1\right)^{-1 / 2} \) |
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Answer» \(\frac d{dx} x(x^2 + 1)^{\frac{-1}2} = x\frac d{dx} (x^2 + 1)^{\frac{-1}2} + \frac{dx}{dx} \times (x^2 + 1)^{\frac{-1}2}\) \( = x\times \frac{-1}2 (x^2 + 1)^{\frac{-1}2-1} \frac{d}{dx} (x^2 + 1) + (x^2 + 1)^{\frac{-1}2}\) \(= -\frac x2 (x^2 + 1)^{\frac{-3}2}.2x + (x^2 + 1)^\frac{-1}2\) \( = (x^2 + 1)^\frac{-1}2 \left(\frac{-x^2}{x^2 + 1}+ 1\right)\) \( = (x^2 + 1)^\frac{-1}2 \left(\frac1{x^2 + 1}\right)\) \( = \frac1{(x^2 + 1)^\frac32}\) |
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