Implicit differentiation of In(x^2 + y^2) = 2arcsin(x/y)

chengeto

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Find dydx\displaystyle \frac{dy}{dx} at an arbtrary point on this curve ln(x2+y2)=2arcsinxy\displaystyle ln(x^2+y^2)\, =\,2arcsin \,\frac{x}{y}

My attempt to solution:

1x2+y22x+(4y2)(dydx)=211(xy)2yxdydxy2\displaystyle \frac{1}{x^2+y^2}2x+(4y^2)(\frac{dy}{dx})=2\frac{1}{\sqrt1-(\frac{x}{y})^2}\frac{y-x\frac{dy}{dx}}{y^2}

Here l get stuck with how to isolate for dy/dx. I also do not know if l differentiated xy\displaystyle \frac{x}{y} correctly.
 
Hello, chengeto!

This is The Implicit Differentiation Problem from **** (coming to a theater near you).


Find dydx for: ln(x2+y2)=2arcsin(xy)\displaystyle \text{Find }\frac{dy}{dx}\,\text{ for: }\:\ln(x^2+y^2)\: =\:2\arcsin\left(\tfrac{x}{y}\right)

. . . . . . . 1x2+y2(2x+2ydydx)  =  211(xy)2yxdydxy2\displaystyle \frac{1}{x^2+y^2}\cdot\left(2x + 2y\,\frac{dy}{dx}\right) \;=\;2\cdot\frac{1}{\sqrt{1 - (\frac{x}{y})^2}}\cdot\frac{y-x\,\frac{dy}{dx}}{y^2}

. . . . . . . . . . . . . . 2(x+ydydx)x2+y2  =  2(yxdydx)y2x2y2y2\displaystyle \frac{2\left(x + y\,\frac{dy}{dx}\right)}{x^2+y^2} \;=\;\frac{2\left(y - x\,\frac{dy}{dx}\right)}{\sqrt{\frac{y^2-x^2}{y^2}}\cdot y^2}

. . . . . . . . . . . . . . . . .x+ydydxx2+y2  =  yxdydxyy2x2\displaystyle \frac{x + y\,\frac{dy}{dx}}{x^2+y^2} \;=\;\frac{y - x\,\frac{dy}{dx}}{y\sqrt{y^2-x^2}}

. . . . . .yy2x2(x+ydydx)  =  (x2+y2)(yxdydx)\displaystyle y\sqrt{y^2-x^2}\cdot\left(x + y\,\frac{dy}{dx}\right) \;=\;(x^2+y^2)\cdot\left(y - x\,\frac{dy}{dx}\right)

. .xyy2x2+y2y2x2dydx  =  y(x2+y2)x(x2+y2)dydx\displaystyle xy\sqrt{y^2-x^2} + y^2\sqrt{y^2-x^2}\,\frac{dy}{dx} \;=\;y(x^2+y^2) - x(x^2+y^2)\,\frac{dy}{dx}

y2y2x2dydx+x(x2+y2)dydx  =  y(x2+y2)xyy2x2\displaystyle y^2\sqrt{y^2-x^2}\,\frac{dy}{dx} + x(x^2+y^2)\,\frac{dy}{dx} \;=\;y(x^2+y^2) - xy\sqrt{y^2-x^2}

. [y2y2x2+x(x2+y2)]dydx  =  y[x2+y2xy2x2]\displaystyle \bigg[y^2\sqrt{y^2-x^2} + x(x^2+y^2)\bigg]\,\frac{dy}{dx} \;=\;y\bigg[x^2+y^2 - x\sqrt{y^2-x^2}\bigg]

. . . . . . . . . . . . . . . . . . . dydx  =  y(x2+y2xy2x2)y2y2x2+x(x2+y2)\displaystyle \frac{dy}{dx} \;=\;\frac{y(x^2+y^2-x\sqrt{y^2-x^2})}{y^2\sqrt{y^2-x^2} + x(x^2+y^2)}

 
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