What is \( \frac{dy}{dx} \) equal to ?
For the next two (02) items that follow : Let \( (e^y)^x - y = 0 \), where y is a function of x whose domain is (0, 10].
- A. \frac{y}{1-xy}
- B. \frac{y}{1+xy}
- C. \frac{y^2}{1-xy} ✓
- D. \frac{y^2}{1+xy}
Correct Answer: C. \frac{y^2}{1-xy}
Explanation
The equation is \( e^{xy} = y \), which gives \( xy = \ln y \). Differentiating both sides with respect to x yields \( y + x \frac{dy}{dx} = \frac{1}{y} \frac{dy}{dx} \). Rearranging for \( \frac{dy}{dx} \) gives \( \frac{y^2}{1-xy} \).
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