For the following demand equation, differentiate implicitly to find $\frac{\mathrm{dp}}{\mathrm{dx}}$. (Hint: Clear the fraction first.) \[ \frac{6 x p}{x+p}=2 \] \[ \frac{d p}{d x}= \]

Step 1 ：First, we clear the fraction in the given equation. We multiply both sides by \(x+p\) to get \(6xp = 2(x+p)\).

Step 2 ：Next, we expand the right side of the equation to get \(6xp = 2x + 2p\).

Step 3 ：We rearrange the equation to isolate \(p\) on one side, giving us \(6xp - 2p = 2x\).

Step 4 ：We factor out \(p\) from the left side of the equation to get \(p(6x - 2) = 2x\).

Step 5 ：We then divide both sides by \(6x - 2\) to solve for \(p\), giving us \(p = \frac{2x}{6x - 2}\).

Step 6 ：Now, we differentiate \(p\) with respect to \(x\) implicitly. Using the quotient rule, we get \(\frac{dp}{dx} = \frac{(6x - 2)(2) - 2x(6)}{(6x - 2)^2}\).

Step 7 ：Simplify the numerator to get \(\frac{dp}{dx} = \frac{12x - 4 - 12x}{(6x - 2)^2}\).

Step 8 ：Further simplifying gives us \(\frac{dp}{dx} = \frac{-4}{(6x - 2)^2}\).

Step 9 ：Finally, we check if our result meets the requirements of the problem. The result is in its simplest form and is a valid derivative of \(p\) with respect to \(x\).

Step 10 ：So, the derivative of \(p\) with respect to \(x\) is \(\frac{dp}{dx} = \frac{-4}{(6x - 2)^2}\).