Vector calculus

Directional Derivative, as it sounds, is the derivative of a function in $E^n$ space in the direction of a specified vector in space.

Or mathematically,

$\displaystyle \nabla_{\overrightarrow{v}}(w) = \nabla(w) \cdot \hat{v}$

$\displaystyle = \nabla(w) \cdot \frac{\overrightarrow{v}}{|\overrightarrow{v}|}$

Now, $w = x^2 + y^2 + 4xz$ , $v = 2\hat{i} - 2\hat{j} - \hat{k}$

$\displaystyle \nabla(w)|_{(1,-2,2)} \cdot \frac{2\hat{i} - 2\hat{j} - \hat{k}}{3}$

And we know that

$\displaystyle \nabla(w) = \frac{\partial(w)}{\partial x} \hat{i} + \frac{\partial(w)}{\partial y} \hat{j} + \frac{\partial(w)}{\partial z} \hat{k}$

$\displaystyle \nabla(w) = 20\hat{i} + 8 \hat{j} - 4\hat{k}$

Coming back to our problem , we substitute and get

$\displaystyle \frac{24}{3} = 8$