What is $(dy)/(dx)$ of $xsqrt(1+y)+ysqrt(1+x)=0??$

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Answer 1 · 2018-02-24T02:16:58

$y' = -(sqrt(1+y) +y/(2sqrt(1+x)))/(x/(2sqrt(1+y)) + sqrt(1+x))$

Given: $x sqrt(1+y) + y sqrt(1+x) = 0$

We need to use implicit differentiation. We will use these differentiation rules for each segment of the equation:

The product rule: $(u*v)' = u*v' + v*u'$

The power rule: $(u^n)' = n u^(n-1) u'$

$(0)' = 0$

First segment of the equation:
Let $u = x; " "u' = 1$
$" "v = sqrt(1+y) = (1+y)^(1/2); " " v' = 1/2(1+y)^(-1/2)y'$
$" "v' = (y')/(2sqrt(1+y)$

Second segment of the equation:
Let $u = y; " "u' = y'$
$" "v = sqrt(1+x) = (1+x)^(1/2); " " v' = 1/2(1+x)^(-1/2)(1)$
$" "v' = 1/(2sqrt(1+x)$

Putting it all together using the product rule:
$(xy')/(2sqrt(1+y)) + sqrt(1+y) + y/(2sqrt(1+x)) +y' sqrt(1+x)= 0$

Keep all terms that contain $y'$ on the left side of the equation:
$(xy')/(2sqrt(1+y)) +y' sqrt(1+x) = -sqrt(1+y) -y/(2sqrt(1+x))$

Factor $y'$ ;

$y' (x/(2sqrt(1+y)) + sqrt(1+x)) = -(sqrt(1+y) +y/(2sqrt(1+x)))$

Isolate $y'$ using division:

$y' = -(sqrt(1+y) +y/(2sqrt(1+x)))/(x/(2sqrt(1+y)) + sqrt(1+x))$

This answer can be simplified if needed by finding common denominators in both the numerator and denominator.

What is (dy)/(dx)(dy)/(dx)of xsqrt(1+y)+ysqrt(1+x)=0??xsqrt(1+y)+ysqrt(1+x)=0??