Multiple Choice 1 point If sin(xy)=x^3 then (dy)/(dx)= (sec(xy))/(3x^2) (3x^2cdot sec(xy)-y)/(x) (3x^2cdot sec(xy))/(y) 3x^2cdot sec(xy)-y

To find $\frac{dy}{dx}$, we need to differentiate both sides of the equation $sin(xy)=x^{3}$ with respect to $x$.<br /><br />Using the chain rule, we have:<br /><br />$\frac{d}{dx}[sin(xy)] = \frac{d}{dx}[x^{3}]$<br /><br />$cos(xy) \cdot \frac{d(xy)}{dx} = 3x^{2}$<br /><br />Now, we need to find $\frac{d(xy)}{dx}$. Using the product rule, we have:<br /><br />$\frac{d(xy)}{dx} = y + x \cdot \frac{dy}{dx}$<br /><br />Substituting this into the previous equation, we get:<br /><br />$cos(xy) \cdot (y + x \cdot \frac{dy}{dx}) = 3x^{2}$<br /><br />Now, we need to solve for $\frac{dy}{dx}$. Rearranging the equation, we have:<br /><br />$cos(xy) \cdot y + cos(xy) \cdot x \cdot \frac{dy}{dx} = 3x^{2}$<br /><br />$cos(xy) \cdot x \cdot \frac{dy}{dx} = 3x^{2} - cos(xy) \cdot y$<br /><br />$\frac{dy}{dx} = \frac{3x^{2} - cos(xy) \cdot y}{cos(xy) \cdot x}$<br /><br />Therefore, the correct answer is $\frac{3x^{2} \cdot sec(xy) - y}{x}$.