Answer
An equation of the tangent plane:
$z = 33x + 8y - 42$
Work Step by Step
We have $f\left( {x,y} \right) = x{y^2} - xy + 3{x^3}y$.
The partial derivatives are
${f_x} = {y^2} - y + 9{x^2}y$, ${\ \ }$ ${f_y} = 2xy - x + 3{x^3}$
Using Theorem 1 of Section 15.4), the equation of the tangent plane to $f\left( {x,y} \right)$ at $P = \left( {1,3} \right)$ is
$z = f\left( {1,3} \right) + {f_x}\left( {1,3} \right)\left( {x - 1} \right) + {f_y}\left( {1,3} \right)\left( {y - 3} \right)$
$z = 15 + 33\left( {x - 1} \right) + 8\left( {y - 3} \right)$
$z = 33x + 8y - 42$
