Question

If $y = y(x)$ is an implicit function of $x$ such that $\log_e (x + y) = 4xy$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is equal to _________.

Hint:

When finding higher derivatives of implicit functions at a specific point, always solve for $y, y', y'', \dots$ sequentially by substituting the numerical values at each step.

Answer 1

Correct Answer: 40

Step 1: Understanding the Concept:
We use implicit differentiation to find the first and second derivatives. First, evaluate $y$ at $x = 0$, then find $y'$, and finally $y''$.
Step 2: Detailed Explanation:
Given $\log_e(x + y) = 4xy$.
At $x = 0$: $\log_e y = 0 \implies y = 1$.
Differentiate with respect to $x$:
\[ \frac{1}{x + y}(1 + y') = 4y + 4xy' \]
At $x = 0, y = 1$:
\[ \frac{1}{1}(1 + y') = 4(1) + 0 \implies 1 + y' = 4 \implies y' = 3 \text{ at } x = 0 \]
Differentiate again:
\[ \frac{(x + y)y'' - (1 + y')^2}{(x + y)^2} = 4y' + 4y' + 4xy'' = 8y' + 4xy'' \]
At $x = 0, y = 1, y' = 3$:
\[ \frac{(1)y'' - (1 + 3)^2}{1^2} = 8(3) + 0 \]
\[ y'' - 16 = 24 \implies y'' = 40 \]
Step 3: Final Answer:
The second derivative at $x = 0$ is 40.