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Question Number 184720 by mustafazaheen last updated on 10/Jan/23

$$$$$$y=\left(\sqrt{3{xy}^2}\right)\left(\sqrt[5]{x^5{y}^2}\right)$$$$y^{\prime }=?y^{{}^{″}}=?$$

Answered by Frix last updated on 10/Jan/23

$$y=\sqrt{3{xy}^2}\times \sqrt[5]{x^5{y}^2}=\sqrt{3x}\mid y\mid \times x\sqrt[5]{y^2}$$$$\mathrm{If}x,y\in \mathbb{R}\Rightarrow x⩾0\wedge y⩾0\Rightarrow$$$$y=\sqrt{3}{x}^{\frac{3}{2}}{y}^{\frac{7}{5}}$$$$\Rightarrow$$$$y=\frac{1}{3x^3\sqrt[4]{3x^3}}$$$$\Rightarrow$$$$y^{\prime }=-\frac{5}{4x^4\sqrt[4]{3x^3}}$$$$y^{″}=\frac{95}{16x^5\sqrt[4]{3x^3}}$$

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