Step 4 Substituting the above back into \( \frac{d}{d x}\left[9 x^{3}+x^{2} y-x y^{3}\right]=\frac{d}{d x} \) [5], we now have the following equation. \[ 27 x^{2}+\left(x^{2} y^{\prime}+2 x y\right)-\left(3 x y^{2} y^{\prime}+y^{3}\right)=0 \] We now rearrange this so that all terms containing \( y^{\prime} \) are on the same side, and get the following. \( 27 x^{2}+2 x y-y^{3}=\left(\square y^{\prime}-x^{2} y^{\prime}\right. \)

Did you know that the mathematical expressions you're working with are a beautiful representation of implicit differentiation? This technique allows us to differentiate equations where \( y \) cannot be isolated easily. It's like solving a puzzle, where you discover the relationship between \( x \) and \( y \) intricately intertwined in just one equation! In practical scenarios, implicit differentiation is invaluable when dealing with curves or relationships that aren't easily expressed as \( y = f(x) \). For example, consider analyzing the trajectory of a rocket launch. The equations of motion might inherently involve both horizontal and vertical components that aren't straightforward to separate. Using implicit differentiation helps engineers calculate rates of change for different forces acting on the rocket, ensuring they design safer and more efficient launches!