The demand $x$ is the number of items that can be sold at a"price of $\$ p$. For $x=\sqrt[3]{6,000-p^{3}}$, find the rate of change of $p$ with respect to $x$ by differentiating implicitly.

Step 1 ：Given the equation \(x=\sqrt[3]{6000-p^{3}}\), we want to find the rate of change of \(p\) with respect to \(x\). This is equivalent to finding \(\frac{dp}{dx}\).

Step 2 ：To do this, we first need to differentiate both sides of the equation with respect to \(x\).

Step 3 ：Starting with the left side, the derivative of \(x\) with respect to \(x\) is simply 1.

Step 4 ：For the right side, we use the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 5 ：The outer function is \(\sqrt[3]{u}\) and the inner function is \(6000-p^{3}\). The derivative of the outer function is \(\frac{1}{3}u^{-2/3}\) and the derivative of the inner function is \(-3p^{2}\).

Step 6 ：Applying the chain rule, we get \(\frac{1}{3}(6000-p^{3})^{-2/3}*(-3p^{2})\).

Step 7 ：Simplifying this expression, we get \(-p^{2}(6000-p^{3})^{-2/3}\).

Step 8 ：Setting the derivatives equal to each other, we get \(1=-p^{2}(6000-p^{3})^{-2/3}\).

Step 9 ：Solving for \(\frac{dp}{dx}\), we get \(\frac{dp}{dx}=-\frac{1}{p^{2}(6000-p^{3})^{2/3}}\).

Step 10 ：Thus, the rate of change of \(p\) with respect to \(x\) is \(\boxed{-\frac{1}{p^{2}(6000-p^{3})^{2/3}}}\).