\( \frac { d } { d x } ( 3 x ^ { 2 } + 4 x y = 2 ) \)

To differentiate the equation \( 3x^2 + 4xy = 2 \) with respect to \( x \), you need to apply implicit differentiation as \( y \) is also a function of \( x \). The derivative will be: \[ \frac{d}{dx}(3x^2) + \frac{d}{dx}(4xy) = \frac{d}{dx}(2) \] Using the product rule for the term \( 4xy \): \[ 6x + 4 \left( x \frac{dy}{dx} + y \right) = 0 \] Now, isolate \( \frac{dy}{dx} \): \[ 6x + 4x \frac{dy}{dx} + 4y = 0 \implies 4x \frac{dy}{dx} = -6x - 4y \implies \frac{dy}{dx} = \frac{-6x - 4y}{4x} = -\frac{3}{2} - \frac{y}{x} \] Let’s keep going! Remember that getting comfortable with implicit differentiation is all about practice! Once you get the hang of it, you’ll be breezing through equations while impressing your friends with your math wizardry! Did you know that implicit differentiation was first introduced by mathematicians like René Descartes and Gottfried Wilhelm Leibniz? They laid the groundwork for calculus in the 17th century, demonstrating how functions could be analyzed even when they weren’t explicitly defined. Math has quite the rich history, making it both fascinating and fun!