Find the gradient of \(xy^{2} + x^{2} y = 4xy\) at the point (1, 3).

Answer Details

Differentiate \(xy^{2}+x^{2}y=4xy\) implicitly with respect to \(x\), treating \(y\) as a function of \(x\) and using the product rule on each term.

\[\underbrace{y^{2}+2xy\frac{dy}{dx}}_{\frac{d}{dx}(xy^{2})}+\underbrace{2xy+x^{2}\frac{dy}{dx}}_{\frac{d}{dx}(x^{2}y)}=\underbrace{4y+4x\frac{dy}{dx}}_{\frac{d}{dx}(4xy)}.\]

Collect the \(\dfrac{dy}{dx}\) terms on one side:

\[(2xy+x^{2}-4x)\frac{dy}{dx}=4y-y^{2}-2xy,\]

\[\frac{dy}{dx}=\frac{4y-y^{2}-2xy}{2xy+x^{2}-4x}.\]

Substitute the point \((1,3)\):

\[\text{Numerator}=4(3)-3^{2}-2(1)(3)=12-9-6=-3,\]

\[\text{Denominator}=2(1)(3)+1^{2}-4(1)=6+1-4=3.\]

\[\frac{dy}{dx}=\frac{-3}{3}=-1.\]

The gradient at \((1,3)\) is \(-1\).