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

Expert Answers info

sciencesolve eNotes educator | Certified Educator

calendarEducator since 2011

write5,349 answers

starTop subjects are Math, Science, and Business

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).`

check Approved by eNotes Editorial

Unlock This Answer Now