[tex]y(x)=x \arcsin \dfrac x3+\sqrt{9-x^2}\\y'(x)=\arcsin \dfrac{x}{3}+\dfrac{x}{3\sqrt{1-(x/3)^2}}+\dfrac{-2x}{2\sqrt{9-x^2}}=\\=\arcsin \dfrac x3+\dfrac{x}{3\sqrt{1-(x/3)^2}}-\dfrac{x}{\sqrt{9-x^2}}=\arcsin\dfrac x3+\dfrac{x}{\sqrt{9-x^2}}-\\-\dfrac{x}{\sqrt{9-x^2}}=\arcsin \dfrac x3.\\(x^2+yx^2-x)'=(2x^5+3y)'\\2x+y'x^2+2xy-1=10x^4+3y'\\y'x^2-3y'=10x^4-2x-2xy+1\\y'(x^2-3)=10x^4-2x-2xy+1\\y'=\dfrac{10x^4-2x-2xy+1}{x^2-3}[/tex]

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[tex]y'(x)=\dfrac{(\ln \sin t)'}{(\sin t)'}=\dfrac{\cos t}{\sin t}:\cos t=\dfrac{1}{\sin t}.[/tex]