Answer
$\dfrac{3}{4}, \lt \dfrac{1}{4},-\dfrac{1}{2},-\dfrac{1}{2} \gt$
Work Step by Step
Formula to calculate the maximum rate of change of $f$: $D_uf=|\nabla f(x,y)|$
$\nabla f(x,y)=\lt 1/y+z , -x/(y+z)^2,-x/(y+z)^2 \gt $
$\nabla f(8,1,3)=\lt 1/1+3 , -8/(1+3)^2,-8/(1+3)^2 \gt=\lt \dfrac{1}{4},-\dfrac{1}{2},-\dfrac{1}{2} \gt$
$|\nabla f(8,1,3)|=\sqrt{(\dfrac{1}{4})^2+(-\dfrac{1}{2})^2+(-\dfrac{1}{2})^2}=\dfrac{3}{4}$
Hence, the required answers are:
$\dfrac{3}{4}, \lt \dfrac{1}{4},-\dfrac{1}{2},-\dfrac{1}{2} \gt$
