How do you implicitly differentiate $-y=xy-e^ysqrt(x-2)$ ?

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Answer 1 · 2016-09-05T12:56:05

$dy/dx=(y(x-5))/{2(x+1)(x-2)(y-1)}$ .

Let us rewrite the given eqn. as, $: e^ysqrt(x-2)=xy+y=y(x+1)$ .

$:. ln(e^ysqrt(x-2))=ln{y(x+1)}$ .

$:. lne^y+ln(x-2)^(1/2)=lny+ln(x+1)$

$:. y+1/2ln(x-2)=lny+ln(x+1)$ .

$:. d/dx{y+1/2ln(x-2)}=d/dx{lny+ln(x+1)}$ .

$:. dy/dx+1/2d/dxln(x-2)=d/dx(lny)+d/dxln(x+1)$ .

$:. dy/dx+1/2*1/(x-2)=d/dylny*dy/dx+1/(x+1)......"[Chain Rule]"$ .

$:. dy/dx+1/(2(x-2))=1/y*dy/dx+1/(x+1$ .

$:. (1-1/y)dy/dx=1/(x+1)-1/(2(x-2))$ .

$:. (y-1)/y*dy/dx={2(x-2)-(x+1)}/{2(x+1)(x-2)}$ .

$:. dy/dx=(y(x-5))/{2(x+1)(x-2)(y-1)}$ .

Enjoy Maths.!

How do you implicitly differentiate -y=xy-e^ysqrt(x-2) -y=xy-e^ysqrt(x-2) ?