Find dy/dx in terms of x and y when x^3+xy^2=5x

1. Differentiate \(x^3\) with respect to \(x\):
\frac{d}{dx}(x^3) = 3x^2

2. Differentiate \(xy^2\) using the product rule:
\frac{d}{dx}(xy^2) = y^2 + 2xy \frac{dy}{dx}

3. Differentiate \(5x\) with respect to \(x\):
\frac{d}{dx}(5x) = 5

Combine all terms:
3x^2 + y^2 + 2xy \frac{dy}{dx} = 5

Solve for \(\frac{dy}{dx}\):
2xy \frac{dy}{dx} = 5 - 3x^2 - y^2
\frac{dy}{dx} = \frac{5 - 3x^2 - y^2}{2xy}

Answer:

\frac{dy}{dx} = \frac{5 - 3x^2 - y^2}{2xy}