Find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for $f(x, y)=6 x^{3}+8 y^{2}-4 x+7 y$ and evaluate each at $(3,4)$
\[
\begin{array}{l}
\frac{\partial f}{\partial x}= \\
\left.\frac{\partial f}{\partial x}\right|_{(3,4)}= \\
\frac{\partial f}{\partial y}= \\
\left.\frac{\partial f}{\partial y}\right|_{(3,4)}=
\end{array}
\]

Step 1 ：Differentiate the function \(f(x, y)=6 x^{3}+8 y^{2}-4 x+7 y\) with respect to \(x\) treating \(y\) as a constant to get \(\frac{\partial f}{\partial x} = 18x^{2} - 4\).

Step 2 ：Differentiate the function \(f(x, y)=6 x^{3}+8 y^{2}-4 x+7 y\) with respect to \(y\) treating \(x\) as a constant to get \(\frac{\partial f}{\partial y} = 16y + 7\).

Step 3 ：Evaluate \(\frac{\partial f}{\partial x}\) at the point \((3,4)\) to get \(\left.\frac{\partial f}{\partial x}\right|_{(3,4)} = 18(3)^{2} - 4 = 158\).

Step 4 ：Evaluate \(\frac{\partial f}{\partial y}\) at the point \((3,4)\) to get \(\left.\frac{\partial f}{\partial y}\right|_{(3,4)} = 16(4) + 7 = 71\).

Step 5 ：So, the final results are \(\boxed{\frac{\partial f}{\partial x} = 18x^{2} - 4, \left.\frac{\partial f}{\partial x}\right|_{(3,4)} = 158}\) and \(\boxed{\frac{\partial f}{\partial y} = 16y + 7, \left.\frac{\partial f}{\partial y}\right|_{(3,4)} = 71}\).