Differentiate 2(x^2+y^2) = 25(x^2−y^2) using implicit differentiation  How would I solve this using implicit differentiation?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use implicit differentiation, hence, you need to differentiate both sides with respect to x, using chain rule such that:

`(d(2(x^2+y^2)))/(dx) = (d(25(x^2-y^2)))/(dx)`

`2*(2x + 2y*(dy)/(dx)) = 25(2x - 2y*(dy)/(dx))`

You need to open the brackets such that:

`4x + 4y*(dy)/(dx) = 50x - 50y*(dy)/(dx)`

You need to isolate to the left the terms that contain `(dy)/(dx)`  such that:

`4y*(dy)/(dx) + 50y*(dy)/(dx) = 50x - 4x`

You need to factor out `(dy)/(dx)`  such that:

`(dy)/(dx)(4y + 50y) = 46x`

`(dy)/(dx) = (46x)/(54y) => (dy)/(dx) = (23x)/(27y)`

You need to find y using the given expression `2(x^2+y^2) = 25(x^2-y^2)`  such that:

`2(x^2+y^2) = 25(x^2-y^2) => 2x^2 + 2y^2 = 25x^2 - 25y^2`

You need to isolate the terms that contain y to the left side such that:

`2y^2 + 25y^2 =25x^2 - 2x^2`

`27y^2 = 23x^2 => y^2 = (23x^2)/27 => y = +-xsqrt(23/27)`

`(dy)/(dx) = (23x)/(27y) => (dy)/(dx) = (23x)/(27(+-xsqrt(23/27)))`

`(dy)/(dx) = +-23sqrt27/(27sqrt23)`

Hence, evaluating `(dy)/(dx)`  using implicit differentiation yields `(dy)/(dx) = +-23sqrt27/(27sqrt23).`

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