# Thread: Directional derivative

1. ## Directional derivative

Find $\displaystyle a,b,c$ constants so that the directional derivative of $\displaystyle f(x,y,z)=axy^2+byz+cz^2x^3$ on $\displaystyle (1,2,-1)$ has a maximum value of $\displaystyle 64$ on a parallel direction to the $\displaystyle z$ axis.

I think we can calculate the directional derivative by using $\displaystyle \langle\nabla f(x_0),x_0\rangle$ where $\displaystyle x_0=(1,2,-1),$ but a maximum value is asked which I don't get, and I don't either get when it says "on a parallel direction to the $\displaystyle z.$"

2. Use:

$\displaystyle \nabla f(1,2,-1)=k(0,0,1)$

so,

$\displaystyle \left\|{\nabla f(1,2,-1)}\right\|=|k|=64$

Fernando Revilla

3. Originally Posted by FernandoRevilla
Use:

$\displaystyle \nabla f(1,2,-1)=k(0,0,1)$

so,

$\displaystyle \left\|{\nabla f(1,2,-1)}\right\|=|k|=64$
Okay but, that would imply that $\displaystyle k=64$ ?

Then how to get $\displaystyle a,b,c$?

Thanks!

4. The directional derivative to the direction to the axis is

$\displaystyle \nabla f \cdot (0,0,1)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})\cdot (0,0,1)=\frac{\partial f}{\partial z} \; | \; _{at \; point \; (1,2,-1)}=64.$

5. The deriviative in the direction of the z axis is simply the partial derivative with respect to z, $\displaystyle by+ 2czx^4$, evaluated at (1, 2, -1). That will give you as single equation for b and c. a can be anything.