I've been trying to solve this problem but I can't find the solution to it. The problem is as follows,
Calculate the directional derivative of the function: $$ f(x,y) = 3xy^2+2x^2-5x $$ as function of the variables $u$ and $v$: \begin{align} u&=3x^3y+5y^2+2\\ v&=-4-2yx^2+3y \end{align} in the point $$P(x,y)=(-1,2)$$ and the direction of the vector $$w(u,v)=(3,-4)$$
If you could help me I'd appreciate it. Thank you!
The basic thing you need is $f_u$ and $f_v.$ I'll describe how to get the first. The second is like unto it.
So, we have that $$f_u=\frac{f_x}{u_x}+\frac{f_y}{u_y}.$$ Similarly, obtain $f_v.$ Then the gradient is $(f_u,f_v).$ Thus, at the point $(x,y)=(-1,2),$ the gradient is given by $\nabla_{(-1,2)}=(f_u(-1,2),f_v(-1,2)),$ so that in the direction $(3,-4),$ the derivative is given by the dot product of the unit vector $$\frac15(3,-4)$$ and $$\nabla_{(-1,2)}.$$