Implicit Differentiation - Exercise 1

Exercise 1. Find y' if xy³ + x²y² + 3x² - 6 = 1.

Answer. Let us use implicit differentiation. Then we take the derivative of the equation treating y as a function of x. We get

$$(y^3 + x 3 y^2 y') + (2xy^2 + x^2 2 y y') + 6x = 0\;.$$

Algebraic manipulations give

$$y' = - \frac{6x + y^3 + 2x y^2}{3xy^2 + 2x^2y} \cdot$$

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