If \(x^{3}y^{2} = (x + y)^{5}\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{y}{x}\)
Taking logarithms of the absolute values of both sides:
\(3\log_e|x| + 2\log_e|y| = 5\log_e|x + y|\)
Differentiating implicitly with respect to x:
\( \displaystyle \frac{3}{x} + \frac{2}{y}\frac{dy}{dx} = \frac{5}{x + y}\left(1 + \frac{dy}{dx}\right) \)
Collecting the \(\frac{dy}{dx}\) terms:
\( \displaystyle \frac{dy}{dx}\left(\frac{2}{y} - \frac{5}{x + y}\right) = \frac{5}{x + y} - \frac{3}{x} \)
\( \displaystyle \frac{dy}{dx} \cdot \frac{2x - 3y}{y(x + y)} = \frac{2x - 3y}{x(x + y)} \)
\( \displaystyle \frac{dy}{dx} = \frac{y}{x} \)