Whenever you see $e$ raised to a log power, always simplify first. It turns a potentially difficult chain rule problem into a basic power rule problem in seconds.
Step 1: Simplify the expression using Log Properties: Use the power rule of logarithms: $n \log a = \log(a^n)$.
$$3 \log x = \log(x^3)$$
Now substitute this back into the original expression:
$$e^{3 \log x} = e^{\log(x^3)}$$
Step 2: Use the Inverse Property of $e$ and $\log$: Since $e$ and the natural logarithm ($\log$ or $\ln$) are inverse functions, $e^{\log f(x)} = f(x)$.
$$e^{\log(x^3)} = x^3$$
Step 3: Differentiate the Simplified Expression: Now, find the derivative of $x^3$ with respect to $x$ using the power rule:
$$\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2$$
By simplifying first, we avoid the complex application of the chain rule.