I have this practice problem before a test. Use implicit differentiation to find d y / d x for the equation x 3 + y 3 = 3 x y .I have no idea how to do this, I didn't understand my lecturer. Can you guys show me the steps?

Question

I have this practice problem before a test. Use implicit differentiation to find   d  y      /    d  x for the equation      x    3    +      y    3    =  3  x  y  .I have no idea how to do this, I didn&#039;t understand my lecturer. Can you guys show me the steps?

Caylee Davenport · Accepted Answer

Think of   y as a function of   x. I will explicitly write this as   y  (  x  ). Then proceed to differentiate everything with respect to   x as normal, remembering the chain rule.We need to differentiate       x    3    +      y    3    (  x  )  =  3  x  y  (  x  ). Let&#039;s do each term one by one.1. Differentiate       x    3  . You should quickly see this is   3      x    2  .2. To differentiate   (  y  (  x  )      )    3  , we need to remember the chain rule. This can be written in many different ways, but this is a composition of the functions   (  ⋅      )    3    ∘  y  ∘  x. The derivative is   3  (  y  (  x  )      )    2        y    ′    (  x  )3. To differentiate   3  x  y  (  x  ), you must remember the product rule. The derivative is   3  y  ∗  (  x  )  +  3  x      y    ′    (  x  ).So in total, differentiation yields  3      x    2    +  3      y    2    (  x  )      y    ′    (  x  )  =  3  y  (  x  )  +  3  x      y    ′    (  x  )  .In this form, to find       y    ′  , you isolate it (if possible) in this equation. Here, this simplifies (after cancelling factors of 3) to      y    ′    (  x  )  =            y      (      x      )      −              x        2                            y        2            (      x      )      −      x        .

Use implicit differentiation to find dy/dx for the equation x^3+y^3=3xy.

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