Find by `dy/dx` implicit differentiation `x^2-4xy+y^2=4`

Wednesday, November 10, 2010

We are asked to find `(dy)/(dx) ` if ` x^2-4xy+y^2=4 `

Note that this is difficult to write as a function of x, so we take the derivative implicitly:

Working term by term:

`d/(dx)( x^2)=2x `

`d/(dx)( -4xy)=-4y-4x(dy)/(dx) ` using the product rule

`d/(dx)(y^2)=2y(dy)/(dx) `

So we get `2x-4y-4x(dy)/(dx)+2y(dy)/(dx)=0 `

`(dy)/(dx)(2y-4x)=4y-2x `

`(dy)/(dx)=(4y-2x)/(2y-4x)=(2y-x)/(y-2x) `

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

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