To find the directional derivative, first we calculate the gradient, and then we take the dot product of the gradient with the vector we want our directional derivative in.
1.Find the gradient of f(x,y)
∇f=<∂f/∂x, ∂f/∂y> =<4cos(4x+3y), 3cos(4x+3y)>
at (x,y)=(-2, -3), ∇f=<4cos(-17), 3cos(-17)>
2.Find the vector for the direction we're moving in
We know for a unit vector r, the x component is equal to cosΘ and the y component is equal to sinΘ, so r=<sqrt(2)/2, sqrt(2)/2>
3.Now we just have to take the dot product
∇f•r=4cos(-17)*2sqrt(2)/2 + 3cos(-17)*sqrt(2)/2 which simplifies to 7*(sqrt(2)/2)*cos17
