Let for . Which of the following equals…

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

Problem

Let y=(x2+1)sin⁡x x3cos⁡xy=(x^{2}+1)^{\sin x}\,x^{3\cos x} for x>0x>0. Which of the following equals 1ydydx\dfrac{1}{y}\dfrac{dy}{dx}? (a) cos⁡x ln⁡(x2+1)+sin⁡xx2+1−3sin⁡x ln⁡x+3cos⁡xx\cos x\,\ln(x^{2}+1)+\dfrac{\sin x}{x^{2}+1}-3\sin x\,\ln x+\dfrac{3\cos x}{x} (b) cos⁡x ln⁡(x2+1)+2xsin⁡xx2+1−3sin⁡x ln⁡x+3cos⁡xx\cos x\,\ln(x^{2}+1)+\dfrac{2x\sin x}{x^{2}+1}-3\sin x\,\ln x+\dfrac{3\cos x}{x} (c) cos⁡x ln⁡(x2+1)+2xsin⁡xx2+1+3sin⁡x ln⁡x+3cos⁡xx\cos x\,\ln(x^{2}+1)+\dfrac{2x\sin x}{x^{2}+1}+3\sin x\,\ln x+\dfrac{3\cos x}{x} (d) cos⁡x ln⁡(x2+1)+2xsin⁡xx2+1−3sin⁡x ln⁡x−3cos⁡xx\cos x\,\ln(x^{2}+1)+\dfrac{2x\sin x}{x^{2}+1}-3\sin x\,\ln x-\dfrac{3\cos x}{x}

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