Use Implicit differentiation to prove y ″ = − 9 y 2 if 4 x 2 − 2 y 2 = 9

Question

Use Implicit differentiation to prove      y    ″    =            −      9              y      2       if   4      x    2    −  2      y    2    =  9

Kaitlyn Levine · Accepted Answer

differentiate both sides of:   4      x    2    −  2      y    2    =  9      d          d      x        (  4      x    2    −  2      y    2    =  9  )    8  x  −  4  y      y    ′    =  0If you don&#039;t understand why       d          d      x            y    2    =  2  y      y    ′   then you need to talk to your teacher / TA / tutor to explain the concept of implicit differentiation again. If that is the case don&#039;t stress, it is a little slippery.isolate       y    ′        y    ′    =            2      x        y  Now differentiate again (with respect to x) to find       y    ″        y    ″    =            2      y      −      2      x              y        ′                    y      2      and substitute the value of       y    ′   found above.      y    ″    =            2      y      −      2      x                        2          x                y                    y      2              y    ″    =            2              y        2            −      4              x        2                    y      3      and   2      y    2    −  4      x    2    =  −  9 from the original constraint.      y    ″    =            −      9              y      3

Sara Solomon · Accepted Answer

assuming that  y  =  y  (  x  )is given then we get  8  x  −  4  y      y    ′    =  0

Use Implicit differentiation to prove y′′=(−9)/(y^2) if 4x^2−2y^2=9.

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