Answer
$1$
Work Step by Step
The directional derivative of f where f is a function of x and y is denoted by $D_{u}f(x,y)$ and is found by the formula:
$D_{u}f(x,y)=f_{x}(x,y)cosB+f_{y}(x,y)sinB$
Where $cosB$ and $sinB$ are found from the directional unit vector
$u=cosBi+sinBj$
$f(x,y)=3x-4xy+9y$
Our directional unit vector is given.
$v=\frac{3}{5}i+\frac{4}{5}j$
In this case, $cosB=3/5$ and $sinB=4/5$
Now we use the formula to find the directional derivative.
$D_{u}f(x,y)=(3-4y)(3/5)+(-4x+9)(4/5)$
Substituting in the point $(1,2)$ we have
$D_{u}f(1,2)=(3-4(2))(3/5)+(-4(1)+9)(4/5)$
$=1$