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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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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