Question

Find \( \frac{dy}{dx} \), if \( x^3 + 2x^2 y - 3xy^2 + y^3 = 100 \)

Hint:

For implicit differentiation, differentiate both sides of the equation with respect to \( x \), and collect terms involving \( \frac{dy}{dx} \).

Answer 1

Step 1: Implicit differentiation.
We are given the equation \( x^3 + 2x^2 y - 3xy^2 + y^3 = 100 \). To find \( \frac{dy}{dx} \), we differentiate implicitly with respect to \( x \). Differentiating each term: \[ \frac{d}{dx}(x^3) = 3x^2, \quad \frac{d}{dx}(2x^2 y) = 2x^2 \frac{dy}{dx} + 4xy \] \[ \frac{d}{dx}(-3xy^2) = -3 \left( y^2 + 2xy \frac{dy}{dx} \right), \quad \frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx} \] Step 2: Substituting into the equation.
Substitute the derivatives into the equation: \[ 3x^2 + \left( 2x^2 \frac{dy}{dx} + 4xy \right) - 3 \left( y^2 + 2xy \frac{dy}{dx} \right) + 3y^2 \frac{dy}{dx} = 0 \] Step 3: Collecting terms involving \( \frac{dy}{dx} \).
Group the terms with \( \frac{dy}{dx} \) together: \[ 2x^2 \frac{dy}{dx} - 6xy \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = -3x^2 - 4xy + 3y^2 \] Factor out \( \frac{dy}{dx} \): \[ \frac{dy}{dx} \left( 2x^2 - 6xy + 3y^2 \right) = -3x^2 - 4xy + 3y^2 \] Step 4: Solving for \( \frac{dy}{dx} \).
Solving for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{-3x^2 - 4xy + 3y^2}{2x^2 - 6xy + 3y^2} \] Step 5: Conclusion.
Thus, the derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = \frac{-3x^2 - 4xy + 3y^2}{2x^2 - 6xy + 3y^2} \]