I need help finding the implicit differentiation of x 4 + sin ⁡ y = x 3 y 2 I especially need help finding d d x for both sides. I'm having trouble with the product rule on the right side with d y d x of x 3 y 2 .

Question

I need help finding the implicit differentiation of       x    4    +  sin  ⁡  y  =      x    3        y    2   I especially need help finding                     d                              d                x             for both sides. I&#039;m having trouble with the product rule on the right side with                               d                y                              d                x             of       x    3        y    2  .

Jaruckigh · Accepted Answer

I think it helps if you think of   y as a function of   x, and in fact write   y  =  f  (  x  ). Then the product rule says      d          d      x        (      x    3        y    2    )  =      d          d      x        (      x    3    f  (  x      )    2    )  =      d          d      x        (      x    3    )  ⋅  f  (  x      )    2    +      x    3    ⋅      d          d      x        (  f  (  x      )    2    )In order to continue, you need to calculate       d          d      x        (  f  (  x      )    2    ) using the chain rule, and the result is   2  ⋅  f  (  x  )  ⋅      d          d      x        (  f  (  x  )  ). Hence,      d          d      x        (      x    3        y    2    )  =      d          d      x        (      x    3    )  ⋅  f  (  x      )    2    +      x    3    ⋅      d          d      x        (  f  (  x      )    2    )  =  3      x    2    f  (  x      )    2    +      x    3    ⋅  2  ⋅  f  (  x  )  ⋅      d          d      x        (  f  (  x  )  )  .Going back to the notation   y  =  f  (  x  ), and             d      y              d      x        =      d          d      x        (  f  (  x  )  ), we obtain      d          d      x        (      x    3        y    2    )  =  3      x    2        y    2    +  2      x    3    y            d      y              d      x        .

sweetymoeyz · Accepted Answer

4      x    3    +  (  cos  ⁡  y  )  .      y    ′    =  3      x    2        y    2    +      x    3    .2  y  .      y    ′  I used       y    ′   instead of   d  y      /    d  x

I need help finding the implicit differentiation of x^4+siny=x^3y^2 I especially need help finding d/(dx) for both sides.

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