How do you implicitly differentiate # 3x^2 - 2-xy + y/(2x) = 11y #?

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Answer 1 · 2015-12-05T03:05:40

Apply implicit differentiation to find
#dy/dx= (12x^3 - y(2x^2 + 1))/(x(2x^2 + 22x^2 - 1))#

Using Implicit Differentiation requires use of the chain rule in general. In this case we will also use the product rule and the quotient rule.

#d/dx(3x^2 - 2 - xy + y/(2x)) = d/dx(11y)#

#=> d/dx3x^2 - d/dx2-d/dxxy + d/dxy/(2x) = d/dx11y#

#=> 6x - 0 - (1*y + xdy/dx) + (2xdy/dx - 2y)/(4x^2) = 11dy/dx#

#=> 6x - y - xdy/dx + 1/(2x)dy/dx - y/(2x^2) = 11dy/dx#

#=> 11dy/dx + xdy/dx - 1/(2x)dy/dx = 6x - y - y/(2x^2)#

#=>dy/dx(11 + x - 1/(2x)) = 6x - y - y/(2x^2)#

#=> dy/dx = (6x - y - y/(2x^2))/(11 + x - 1/(2x))#

#= (12x^3 - y(2x^2 + 1))/(x(2x^2 + 22x^2 - 1))#