1. **State the problem:** Differentiate the function implicitly given by the equation $6xy$ with respect to $x$. 2. **Recall the product rule:** When differentiating a product of two functions, $u(x)$ and $v(x)$, the derivative is given by $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x).$$ 3. **Apply the product rule:** Here, $u = 6x$ and $v = y$, where $y$ is a function of $x$. So, $$\frac{d}{dx}[6xy] = \frac{d}{dx}[6x \cdot y] = 6 \cdot y + 6x \cdot \frac{dy}{dx}.$$ 4. **Interpret the result:** The derivative of $6xy$ with respect to $x$ is $$6y + 6x \frac{dy}{dx}.$$ This expression includes $\frac{dy}{dx}$ because $y$ depends on $x$ implicitly. 5. **Summary:** The implicit derivative of $6xy$ with respect to $x$ is $$6y + 6x \frac{dy}{dx}.$$