For , evaluate…

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

For x>1x>1, evaluate ddx[(2x+1)x(x2−1)x]\dfrac{d}{dx}\bigl[(2x+1)^{\sqrt{x}}(x^{2}-1)^{x}\bigr]. (a) (2x+1)x(x2−1)x[2x2x+1+2x2x2−1](2x+1)^{\sqrt{x}}(x^{2}-1)^{x}\left[\dfrac{2\sqrt{x}}{2x+1}+\dfrac{2x^{2}}{x^{2}-1}\right] (b) (2x+1)x(x2−1)x[ln⁡(2x+1)2x+x2x+1+ln⁡(x2−1)+2x2x2−1](2x+1)^{\sqrt{x}}(x^{2}-1)^{x}\left[\dfrac{\ln(2x+1)}{2\sqrt{x}}+\dfrac{\sqrt{x}}{2x+1}+\ln(x^{2}-1)+\dfrac{2x^{2}}{x^{2}-1}\right] (c) (2x+1)x(x2−1)x[ln⁡(2x+1)2x+2x2x+1+ln⁡(x2−1)+2x2x2−1](2x+1)^{\sqrt{x}}(x^{2}-1)^{x}\left[\dfrac{\ln(2x+1)}{2\sqrt{x}} + \dfrac{2\sqrt{x}}{2x+1} + \ln(x^{2}-1) + \dfrac{2x^{2}}{x^{2}-1}\right] (d) (2x+1)x(x2−1)x[ln⁡(2x+1)2x+2x2x+1+ln⁡(x2−1)+2xx2−1](2x+1)^{\sqrt{x}}(x^{2}-1)^{x}\left[\dfrac{\ln(2x+1)}{2\sqrt{x}}+\dfrac{2\sqrt{x}}{2x+1}+\ln(x^{2}-1)+\dfrac{2x}{x^{2}-1}\right]

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