1. **State the problem:** Find the derivative with respect to $x$ of the function $f(x) = 3^{\ln x}$. 2. **Recall the formula:** For a function of the form $a^{g(x)}$, the derivative is given by $$\frac{d}{dx} a^{g(x)} = a^{g(x)} \ln(a) \cdot g'(x).$$ 3. **Rewrite the function:** Note that $3^{\ln x} = e^{\ln(3) \cdot \ln x}$ because $a^b = e^{b \ln a}$. 4. **Differentiate using the chain rule:** $$\frac{d}{dx} 3^{\ln x} = \frac{d}{dx} e^{\ln(3) \cdot \ln x} = e^{\ln(3) \cdot \ln x} \cdot \frac{d}{dx} (\ln(3) \cdot \ln x).$$ 5. **Calculate the derivative inside:** Since $\ln(3)$ is constant, $$\frac{d}{dx} (\ln(3) \cdot \ln x) = \ln(3) \cdot \frac{d}{dx} \ln x = \ln(3) \cdot \frac{1}{x}.$$ 6. **Substitute back:** $$\frac{d}{dx} 3^{\ln x} = e^{\ln(3) \cdot \ln x} \cdot \frac{\ln(3)}{x} = 3^{\ln x} \cdot \frac{\ln(3)}{x}.$$ **Final answer:** $$\boxed{\frac{d}{dx} 3^{\ln x} = \frac{3^{\ln x} \ln(3)}{x}}.$$