How do you find #dy/dx# by implicit differentiation given #y=sqrt(xy+1)#?

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Answer 1 · 2017-02-21T02:21:35

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

Square both sides to eliminate the need to use the chain rule.

#y^2 = (sqrt(xy + 1))^2#

#y^2 = xy + 1#

Now use implicit differentiation and the product rule to find the derivative.

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

#2y(dy/dx) - x(dy/dx) = y#

#dy/dx(2y - x) = y#

#dy/dx= y/(2y - x)#

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

Hopefully this helps!