Question

$\frac{d}{dx} \left( e^{3 \log x} \right) =$

• A. $\log x$
• B. $3x$
• C. $x^3$
• D. $3x^2$

Hint:

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.

Answer 1

The Correct Option is D

To solve this derivative easily, we should first simplify the expression using the properties of logarithms and exponents. 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.