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Question Number 38252 by Rio Mike last updated on 23/Jun/18

$$if{x}^2+3xy-y^2=3find$$$$\frac{dy}{dx}atpoint(1,1)hence$$$$differentiate\frac{sinx}{1+x}withrespect$$$$tox.$$

Commented by math khazana by abdo last updated on 23/Jun/18

$$x^2+3xy-y^2=3\Rightarrow y^2-3xy-x^2+3=0$$$$\mathrm{\Delta }=9x^2-4(3-x^2)=13x^2-12soif{x}^2⩾\frac{12}{13}$$$$y\left(x\right)=\frac{3x\stackrel{-}{+}\sqrt{13x^2-12}}{2}\Rightarrow$$$$y^{{}^{\prime }}\left(x\right)=\frac{3}{2}\stackrel{-}{+}\frac{1}{2}\frac{26x}{2\sqrt{13x^2-12}}$$$$=\frac{3}{2}\stackrel{-}{+}\frac{13}{2\sqrt{{13}^{}{x}^2-12}}\Rightarrow$$$$y^{{}^{\prime }}\left(1\right)=\frac{3}{2}\stackrel{-}{+}\frac{13}{2}\Rightarrow y^{{}^{\prime }}\left(1\right)=8or{y}^{{}^{\prime }}\left(1\right)=-5$$

Commented by math khazana by abdo last updated on 23/Jun/18

$$letf\left(x\right)=\frac{sinx}{1+x}soifx\ne -1$$$$f^{{}^{\prime }}\left(x\right)=\frac{cosx(1+x)-sinx}{{(1+x)}^2}=\frac{cosx+xcosx-sinx}{{(1+x)}^2}$$

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