The directional derivative of the scalar function $\log(x^2+y^2+z^2)$ at point P(1,1,1) in direction of line joining p to $p_0(3,2,1)$ is
parametric equation of line joining p to $p_0$ is $\overrightarrow{v}=(2t+1)i+(t+1)j+k$
unit vector in direction of $\overrightarrow{v}$=$\frac{(2t+1)i+(t+1)j+k}{\sqrt{5t^2+6t+3}}$
$\nabla{f} at (1,1,1)=\frac{2}{3}(i+j+k)$
${D^f}_\overrightarrow{v}$=$\nabla{f}.\overrightarrow{v}$
but how to take dot product at $(1,1,1)$?
The directional derivative for $f(x,y,z)$ in the direction $\vec v$ is computed as
$$ \left <\nabla f, \vec v \right > $$
with $\vec v = (v_x,y_y,v_z)$ Here
$$ \nabla f = 2\left(\frac{(x,y,z)}{x^2+y^2+z^2}\right) $$
and
$$ \left <\nabla f, \vec v \right > = 2\left(\frac{x v_x+y v_y+z v_z}{x^2+y^2+z^2}\right) $$
and at point $(x_0,y_0,y_0)$
$$ 2\left(\frac{x_0 v_x+y_0 v_y+z_0 v_z}{x^2_0+y^2_0+z^2_0}\right) = 4 $$