How can I solve the problem?

How to find#dy/(dx)#of the mentioned equation?
#xsqrt(1+y)+ysqrt(1+x)=0#

1 Answer
Jim G.
Dec 28, 2017

#dy/dx=-(y/sqrt(1+x)+2sqrt(1+y))/(x/sqrt(1+y)+2sqrt(1+x))#

Explanation:

#"differentiate "color(blue)"implicitly with respect to x"#

#"differentiate "xsqrt(1+y)" and "ysqrt(1+x)" using "#
#"the "color(blue)"product rule"#

#rArrx(1+y)^(1/2)+y(1+x)^(1/2)=0#

#rArr(x . 1/2(1+y)^(-1/2).dy/dx+(1+y)^(1/2))#

#+(y . 1/2(1+x)^(-1/2)+(1+x)^(1/2).dy/dx)=0#

#rArr1/2x(1+y)^(-1/2)dy/dx+(1+x)^(1/2)dy/dx#

#=-1/2y(1+x)^(-1/2)-(1+y)^(1/2)#

#rArrdy/dx(1/2x(1+y)^(-1/2)+(1+x)^(1/2))#

#=-1/2(y(1+x)^(-1/2)+2(1+y)^(1/2))#

#rArrdy/dx=-(cancel(1/2)(y(1+x)^(-1/2)+2(1+y)^(1/2)))/(cancel(1/2)(x(1+y)^(-1/2)+2(1+x)^(1/2))#

#color(white)(rArrdy/dx)=-(y/sqrt(1+x)+2sqrt(1+y))/(x/sqrt(1+y)+2sqrt(1+x))#

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