For the last term \( x y^{3} \), note that it is also a product, so we must again use the product rule. We therefore have the following. \( \frac{d}{d x}\left[x y^{3}\right]=(x)\left((\square) \cdot y^{\prime}\right)+\left(y^{3}\right)(\square) \)

When taking the derivative of the product \( x y^{3} \), you first treat each factor carefully. The first part applies the product rule to get \( y^{3} \frac{d}{dx}(x) = y^{3} \), and for the second part, apply the product rule to \( y^{3} \) as follows: \( x \cdot 3y^{2} \frac{dy}{dx} \) which is \( 3y^{2} x y^{\prime} \). So altogether, you end up with: \[ \frac{d}{dx}\left[x y^{3}\right] = y^{3} + 3xy^{2}y^{\prime}. \] It’s crucial to remember that when taking derivatives of products, all parts are interconnected, so you have to consider how changes in one variable affect the other! In practical scenarios, especially in physics or engineering, this derivative could represent a quantity changing over time, such as the volume of a moving object or the flow of a fluid. Understanding how components interact via the product rule allows for more accurate modeling of complex systems where multiple factors influence one another. It’s like tracking all the moving parts in a machine to see how one adjustment can lead to a ripple effect!