Mathematics

Mathematics

Problem

For x>0x>0, let y(x)=(x2+3)(ln⁡x)x cos⁡xy(x)=(x^{2}+3)(\ln x)\sqrt{x}\,\cos x. Which of the following expressions equals y′(x)y'(x)? a) 2x(ln⁡x)xcos⁡x+(x2+3)xcos⁡xx+(x2+3)(ln⁡x)cos⁡x2x+(x2+3)(ln⁡x)xsin⁡x2x(\ln x)\sqrt{x}\cos x+\dfrac{(x^{2}+3)\sqrt{x}\cos x}{x}+\dfrac{(x^{2}+3)(\ln x)\cos x}{2\sqrt{x}}+(x^{2}+3)(\ln x)\sqrt{x}\sin x b) 2x(ln⁡x)xcos⁡x+(x2+3)xcos⁡xln⁡x+(x2+3)(ln⁡x)cos⁡x2x−(x2+3)(ln⁡x)xsin⁡x2x(\ln x)\sqrt{x}\cos x+\dfrac{(x^{2}+3)\sqrt{x}\cos x}{\ln x}+\dfrac{(x^{2}+3)(\ln x)\cos x}{2\sqrt{x}}-(x^{2}+3)(\ln x)\sqrt{x}\sin x c) 2x(ln⁡x)xcos⁡x+(x2+3)xcos⁡xx+(x2+3)(ln⁡x)cos⁡x2x−(x2+3)(ln⁡x)xsin⁡x2x(\ln x)\sqrt{x}\cos x+\dfrac{(x^{2}+3)\sqrt{x}\cos x}{x}+\dfrac{(x^{2}+3)(\ln x)\cos x}{2\sqrt{x}}-(x^{2}+3)(\ln x)\sqrt{x}\sin x d) 2x(ln⁡x)xcos⁡x+(x2+3)xcos⁡xx−(x2+3)(ln⁡x)xsin⁡x2x(\ln x)\sqrt{x}\cos x+\dfrac{(x^{2}+3)\sqrt{x}\cos x}{x}-(x^{2}+3)(\ln x)\sqrt{x}\sin x

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