Questions: Use implicit differentiation to find dy/dx. 7 x^2 y + 3 x y^2 = -3 dy/dx = □ □

Solution Steps

To find \(\frac{dy}{dx}\) using implicit differentiation, follow these steps:

1. Differentiate both sides of the given equation with respect to \(x\), treating \(y\) as a function of \(x\).
2. Apply the product rule where necessary.
3. Collect all terms involving \(\frac{dy}{dx}\) on one side of the equation.
4. Solve for \(\frac{dy}{dx}\).

We start with the equation: \[ 7x^2y + 3xy^2 = -3 \] Differentiating both sides with respect to \(x\) gives: \[ \frac{d}{dx}(7x^2y) + \frac{d}{dx}(3xy^2) = \frac{d}{dx}(-3) \] Using the product rule, we find: \[ \frac{d}{dx}(7x^2y) = 14xy + 7x^2\frac{dy}{dx} \] \[ \frac{d}{dx}(3xy^2) = 3y^2 + 6xy\frac{dy}{dx} \] Thus, the differentiated equation becomes: \[ 14xy + 7x^2\frac{dy}{dx} + 3y^2 + 6xy\frac{dy}{dx} = 0 \]

Rearranging the equation to isolate terms involving \(\frac{dy}{dx}\): \[ 7x^2\frac{dy}{dx} + 6xy\frac{dy}{dx} = -14xy - 3y^2 \] Factoring out \(\frac{dy}{dx}\): \[ \frac{dy}{dx}(7x^2 + 6xy) = -14xy - 3y^2 \]

Now, we can solve for \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{-14xy - 3y^2}{7x^2 + 6xy} \]

Final Answer