AP DSC School Assistant Math Model Test 3

Concept:This problem uses the chain rule and properties of logarithms to simplify the function before differentiating. The key step is to express $y$ entirely in terms of $x$ to find the second derivative directly.Explanation:Step 1: Write $y$ in terms of $x$ using the given $t = e^{2x}$.$y = \log_e (t^2) = 2 \log_e t$Since $\log_e t = \log_e(e^{2x}) = 2x$, we have:$y = 2 \times (2x) = 4x$Step 2: Find the first derivative.$\frac{dy}{dx} = \frac{d}{dx}(4x) = 4$Step 3: Find the second derivative.$\frac{d^2 y}{dx^2} = \frac{d}{dx}(4) = 0$Answer:The second derivative is $0$, which corresponds to option A.