Solve the problem.
-About how much will $f ( x , y , z ) = \ln ( - 7 x + 10 y + 9 z )$ change if the point $( x , y , z )$ moves from $( - 1 , - 8,3 )$ a distance of $\mathrm { ds } = \frac { 1 } { 10 }$ unit in the direction of $12 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$ ?
A) $- \frac { 1 } { 2990 }$
B) $\frac { 7 } { 1495 }$
C) $\frac { 2 } { 1495 }$
D) $\frac { 9 } { 2990 }$

Question

Quizplus · Accepted Answer

The directional derivative is calculated using the gradient of $f$ and the unit vector in the given direction. The gradient is $\nabla f = \left( \frac{-7}{-7x+10y+9z}, \frac{10}{-7x+10y+9z}, \frac{9}{-7x+10y+9z} \right)$. At $(-1, -8, 3)$, $\nabla f = \left( \frac{-7}{-1}, \frac{10}{-1}, \frac{9}{-1} \right)$. The unit vector is $\frac{1}{13}(12, 3, 4)$. The directional derivative is $\nabla f \cdot \frac{1}{13}(12, 3, 4) = \frac{9}{2990}$.

Solve the Problem f(x,y,z)=ln⁡(−7x+10y+9z)f ( x , y , z ) = \ln ( - 7 x + 10 y + 9 z )f(x,y,z)=ln(−7x+10y+9z)

Solve the Problem $f ( x , y , z ) = \ln ( - 7 x + 10 y + 9 z )$