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

To differentiate the equation \( x^2 y^3 = 2x \) with respect to \( x \), you need to apply implicit differentiation. This means treating \( y \) as a function of \( x \). The derivative will yield \( 2xy^3 + 3x^2y^2 \frac{dy}{dx} = 2 \). Rearranging terms, you can isolate \( \frac{dy}{dx} \) to get \( \frac{dy}{dx} = \frac{2 - 2xy^3}{3x^2y^2} \). There you have it—the slope of \( y \) with respect to \( x \)! Understanding implicit differentiation is crucial for solving equations where \( y \) isn't explicitly defined. Real-world applications are plentiful, especially in fields like physics and economics, where relationships between variables often exist without clear functions. For instance, you might encounter scenarios involving rates of change, like speed and acceleration, which rely on these principles. By mastering implicit differentiation, you'll be equipped to tackle complex equations encountered in scientific and mathematical modeling!