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Implicit differentiation shortcut

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I’m helping my daughter with her calculus homework, and we just came across implicit differentiation. It’s been a while since I did calculus but I remembered a shortcut I figured out back in the day.

To (implicitly) differentiate the equation x^3y^5+3x=8y^3+1 the way I was originally taught, I would:

  1. Take the derivative of both sides by x, treating y as a function of x using the chain rule.

    1. Start: x^3y^5+3x=8y^3+1

    2. Differentiate (w/product rule): 5x^3y^4y’+3x^2y^5+3=24y^2y’

  2. Isolate y’:

    1. Start: 5x^3y^4y’+3x^2y^5+3=24y^2y’

    2. Group y’ terms on left: 5x^3y^4y’-24y^2y’=-3x^2y^5-3

    3. Factor: y'(5x^3y^4-24y^2)=-3x^2y^5-3

    4. Divide: y’=(-3x^2y^5-3)/(5x^3y^4-24y^2)

The shortcut I figured out was to move everything to one side, and then the answer is: y’=-(derivative of equation by x as though y were constant)/(derivative of equation by y as though x were constant)

So for the above example, the shortcut is:

  1. Start: x^3y^5+3x=8y^3+1

  2. Move all terms to one side: x^3y^5+3x-8y^3-1=0

  3. Take derivative of #2 by x as though y were constant: 3x^2y^5+3

  4. Take derivative of #2 by y as though x were constant: 5x^3y^4-24y^2

  5. Shortcut! Answer is y’=-(3x^2y^5+3)/(5x^3y^4-24y^2)

You avoid using the chain rule to make y’, the product rule, and also the algebra to isolate y’. The derivatives are easier, you can usually just write the answer.

Some examples: (https://www.cliffsnotes.com/study-guides/calculus/calculus/the-derivative/implicit-differentiation)

x^2y^3-xy=10 => x^2y^3-xy-10 => -(2xy^3-y)/(3x^2y^2-x)

y=sin(x)+cos(y) => y-sin(x)-cos(y) => -(-cos(x))/(1+sin(y)) => cos(x)/(1+sin(y))

x^2+3xy+y^2=-1 => x^2+3xy+y^2+1 => -(2x+3y)/(3x+2y)

x^2+y^2=25 => x^2+y^2-25 => -(2x)/(2y) => -x/y

More complicated examples: (https://tutorial.math.lamar.edu/classes/calci/implicitdiff.aspx)

x^2tan(y)+y^10sec(x)=2x => x^2tan(y)+y^10sec(x)-2x => -(2xtan(y)+y^10sec(x)tan(x)-2)/(x^2sec^2(y)+10y^9sec(x))

e^(2x+3y)=x^2−ln(xy^3) => e^(2x+3y)-x^2+ln(xy^3) => -(2e^(2x+3y)-2x+1/x)/(3e^(2x+3y)+3/y)

It’s been right in every case I’ve tried, but I can’t prove it works. Is this a known thing? Can anyone think of a counterexample?