If $3 x^{2}+5 x+x y=3$ and $y(3)=-13$, find $y^{\prime}(3)$ by implicit differentiation.

Step 1 ：Given the equation $3 x^{2}+5 x+x y=3$ and $y(3)=-13$, we are asked to find $y^{\prime}(3)$.

Step 2 ：First, we need to differentiate the given equation implicitly with respect to $x$. The derivative of $3x^2$ is $6x$, the derivative of $5x$ is $5$, and the derivative of $xy$ is $x \cdot y' + y$ using the product rule. The derivative of $3$ is $0$.

Step 3 ：So, the derivative of the given equation is $6x + 5 + x \cdot y' + y = 0$.

Step 4 ：Rearranging the equation to solve for $y'$, we get $y' = -\frac{6x + 5 + y}{x}$.

Step 5 ：Substitute $x=3$ and $y=-13$ into the equation, we get $y'(3) = -\frac{6 \cdot 3 + 5 - 13}{3}$.

Step 6 ：Simplify the equation, we get $y'(3) = -\frac{18 - 13}{3} = -\frac{5}{3}$.

Step 7 ：So, the derivative of $y$ at $x=3$ is $-\frac{5}{3}$.

Step 8 ：\boxed{y'(3) = -\frac{5}{3}}