Question

If \( y \) is a function of \( x \) and \( \log(x + y) = 2xy \), then the value of \( y'(0) \) is

• A. 2
• B. 0
• C. -1
• D. 1

Hint:

When differentiating implicitly, treat \( y \) as a function of \( x \) and apply the chain rule accordingly.

Answer 1

The Correct Option is B

Step 1: Differentiating the equation.
Given \( \log(x + y) = 2xy \), differentiate both sides implicitly with respect to \( x \): \[ \frac{1}{x + y} \cdot (1 + y') = 2y + 2xy' \] Step 2: Solving for \( y'(0) \).
Substitute \( x = 0 \) and solve for \( y'(0) \), which turns out to be 0. Step 3: Conclusion.
Thus, the correct answer is (B).