I am trying to implicitly differentiate sin ⁡ ( x / y ) = 1 / 2The solution manual saysStep 1. cos ⁡ ( x / y ) ⋅ y − x d y d x y 2 = 0But I don't understand how they arrive at this next part:Step 2. y − x d y d x = 0Is cos ⁡ ( x / y ) = y 2 ?

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I am trying to implicitly differentiate  sin  ⁡  (  x      /    y  )  =  1      /    2The solution manual saysStep 1.  cos  ⁡  (  x      /    y  )  ⋅            y      −      x                        d          y                          d          x                            y      2        =  0But I don&#039;t understand how they arrive at this next part:Step 2.  y  −  x            d      y              d      x        =  0Is   cos  ⁡  (  x      /    y  )  =      y    2  ?

Alden Holder · Accepted Answer

The idea is that when   a  b  =  0, you have that   a  =  0 or   b  =  0 as solutions to the equation (or possibly both). Here, we have             1              y        2              cos  ⁡      (                  x        y              )    =  0 or   y  −  x      y    ′    =  0.Since   y cannot be &quot;infinity&quot;, from the first solution we have that   cos  ⁡      (                  x        y              )    =  0 which says       x    y    =      π    2    ,            3      π        2    ,  ⋯. So really,   y is a linear function of   x. But you can check that none of the above values I gave satisfy   sin  ⁡      (                  x        y              )    =            1      2       (in fact it is equal to   ±  1 for these values). Check this yourself.So that leaves us with   y  −  x      y    ′    =  0. Is it clear now?

Greyson Landry · Accepted Answer

One approach says that the cosine cannot be 0 when the sine is 1/2, so if   cos  ⁡  (  something  )  ⋅  (  something  )  =  0, then the second &quot;something&quot; must be 0.Another approach seems simpler, since it avoids differentiating any trigonometric functions and applying the chain rule to that differentiation. Just observe that      x    y    =  arcsin  ⁡      1    2    .Differentiate the left side using the quotient rule and the chain rule. When you differentiate the right side, you get 0 since the right side is constant, and all references to trigonometric functions disappear.

I am trying to implicitly differentiate sin(x/y)=1/2

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