Calculus/calculus

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Expert: - 4/21/2010


Question
calculate dy/dx when (xy)^3 + y = 3x

Answer
The derivative of (xy)³ is 3(xy)² d(xy)/dx.
The derivative of xy is a product rule, so it is y + xy'.
The derivative of y is y' and the derivative of 3x is 3.

Looking batck at the problem, we get 3(xy)²(y+xy') + y' = 3.
That multiplies out to 3x²y³ + 3x³y²y' + y' = 3.

Subtract the first term from both sides, so that only terms with y' are on the left side.
Note that these terms are the same as 3x³y²y' + y', which is (3x³y² + 1)y'.
Divide both sides by (3x³y² + 1), so all that is left on the left side is y'

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