Use implicit differentiation to find \( \frac{d y}{d x} \) \( 3 x^{2} y+2 x y^{2}=-6 \)

To find \( \frac{dy}{dx} \) using implicit differentiation, we start with the equation \( 3x^2y + 2xy^2 = -6 \). Differentiating both sides with respect to \( x \) gives: \[ \frac{d}{dx}(3x^2y) + \frac{d}{dx}(2xy^2) = \frac{d}{dx}(-6) \] Using the product rule on both terms, we have: \[ 3x^2\frac{dy}{dx} + 6xy + 2y^2 + 2x \frac{dy}{dx} = 0 \] Now, let's group the terms involving \( \frac{dy}{dx} \): \[ (3x^2 + 2x)\frac{dy}{dx} + 6xy + 2y^2 = 0 \] Isolating \( \frac{dy}{dx} \), we find: \[ (3x^2 + 2x)\frac{dy}{dx} = -6xy - 2y^2 \] Finally, we can express \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{-6xy - 2y^2}{3x^2 + 2x} \] And voilà! That's our derivative! For a little historical twist, implicit differentiation has roots in the development of calculus, with contributions from greats like Leibniz and Newton in the 17th century. They laid the groundwork that allows us to derive relationships between variables even when they're not isolated. Now, on the application side, implicit differentiation is super handy in real-world scenarios, like in physics when dealing with curves of motion that depend on multiple changing variables. You can use this method to analyze relationships in any situation where explicit equations aren’t feasible!