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Question Number 70582 by Raphael last updated on 05/Oct/19
Given that y=(4/( (√((x^3 +1))))) show that 2(x^3 +1)(dy/dx)=−3x^2 y.
$$\mathrm{Given}\:\mathrm{that}\:\mathrm{y}=\frac{\mathrm{4}}{\:\sqrt{\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)}}\:\mathrm{show}\:\mathrm{that}\:\mathrm{2}\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=−\mathrm{3x}^{\mathrm{2}} \mathrm{y}. \\ $$
Commented by kaivan.ahmadi last updated on 05/Oct/19
2(x^3 +1)×((−((4×3x^2 )/(2(√(x^3 +1)))))/(x^3 +1))=2×((−2×3x^2 )/( (√(x^3 +1))))=  −3x^2 ×(4/( (√(x^3 +1))))=−3x^2 y
$$\mathrm{2}\left({x}^{\mathrm{3}} +\mathrm{1}\right)×\frac{−\frac{\mathrm{4}×\mathrm{3}{x}^{\mathrm{2}} }{\mathrm{2}\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}}}{{x}^{\mathrm{3}} +\mathrm{1}}=\mathrm{2}×\frac{−\mathrm{2}×\mathrm{3}{x}^{\mathrm{2}} }{\:\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}}= \\ $$$$−\mathrm{3}{x}^{\mathrm{2}} ×\frac{\mathrm{4}}{\:\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}}=−\mathrm{3}{x}^{\mathrm{2}} {y} \\ $$

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