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How do you implicitly differentiate # sqrt(xy) = x - 2y#?

Answer 1

William Abdalla

#y'=(2sqrt(xy)-y)/(x+4sqrt(xy))#

#1/(2sqrt(xy)) [xy'+y]=1-2y'#

#xy'+y =(1-2y')(2sqrt(xy)) #

#xy'+y=2sqrt(xy)-4y'sqrt(xy)#

#xy'+4y'sqrt(xy)=2sqrt(xy)-y#

#y'(x+4sqrt(xy))=2sqrt(xy)-y#

#y'=(2sqrt(xy)-y)/(x+4sqrt(xy))#

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Answer 2

Adam Adkins

To implicitly differentiate sqrt(xy) = x - 2y, first, differentiate both sides of the equation with respect to x. Then, apply the chain rule and product rule where necessary. After differentiation, solve for dy/dx. The result is:

dy/dx = (2y - x) / (2sqrt(xy) - 2)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.