use implicit differentiation to find d y d x in terms of x and y1.) x 3 − x y + y 2 = 42.) y = sin ⁡ ( x y )find d 2 y d x 2 3.) x 2 y 2 − 2 x = 3

Question

use implicit differentiation to find             d      y              d      x       in terms of   x and   y1.)       x    3    −  x  y  +      y    2    =  42.)   y  =  sin  ⁡  (  x  y  )find                     d        2            y              d              x        2            3.)       x    2        y    2    −  2  x  =  3

Jazlene Dickson · Accepted Answer

I&#039;ll show you how to do 1), and you can try the others yourself. Differentiating implicitly, we have      d          d      x        (      x    3    −  x  y  +      y    2    )  =      d          d      x        4or  3      x    2    −  y  −  x      y    ′    +  2  y      y    ′    =  0so that  3      x    2    −  y  =      y    ′    (  x  −  2  y  )and hence      y    ′    =            3              x        2            −      y              x      −      2      y        .

Livia Cardenas · Accepted Answer

And 2,  d  y  =  cos  ⁡  (  x  y  )  ∗  (  y  d  x  +  x  d  y  )  =  y  cos  ⁡  (  x  y  )  d  x  +  x  cos  ⁡  (  x  y  )  d  yusing chain and product rules. Then solve for   d  y, and then finally             d      y              d      x      ,  d  y  −  x  cos  ⁡  (  x  y  )  d  y  =  cos  ⁡  (  x  y  )  ∗  (  y  d  x  +  x  d  y  )  =  y  cos  ⁡  (  x  y  )  d  x  ⇒  d  y  (  1  −  x  cos  ⁡  (  x  y  )  )  =  y  cos  ⁡  (  x  y  )  d  x  ⇒            d      y              d      x        =            y      cos      ⁡      (      x      y      )              1      −      x      cos      ⁡      (      x      y      )

use implicit differentiation to find (dy)/(dx) in terms of x and y 1.) x^3−xy+y^2=4 2.) y=sin(xy) find (d^2y)/(dx^2) 1.) x^2y^2−2x=3

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