Question

If \( y = \sin(xy) \), then find \( \frac{dy}{dx} \):

Hint:

For implicit differentiation, differentiate both sides and use the product and chain rules carefully. Collect \( \frac{dy}{dx} \) terms on one side to solve.

Answer 1

Given: \[ y = \sin(xy) \] Differentiate both sides with respect to \( x \): \[ \frac{dy}{dx} = \cos(xy) \cdot \frac{d}{dx}(xy) \] Using product rule on \( xy \): \[ \frac{d}{dx}(xy) = y + x \frac{dy}{dx} \] So, \[ \frac{dy}{dx} = \cos(xy) \cdot \left( y + x \frac{dy}{dx} \right) \] Rearranging to solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = y \cos(xy) + x \cos(xy) \frac{dy}{dx} \] Bring terms involving \( \frac{dy}{dx} \) to one side: \[ \frac{dy}{dx} - x \cos(xy) \frac{dy}{dx} = y \cos(xy) \] \[ \frac{dy}{dx} (1 - x \cos(xy)) = y \cos(xy) \] Therefore, \[ \boxed{ \frac{dy}{dx} = \frac{y \cos(xy)}{1 - x \cos(xy)} } \]