Find the derivative of the function at P0 in the direction of u.
-$f ( x , y , z ) = \ln \left( x ^ { 2 } + 9 y ^ { 2 } - 7 z ^ { 2 } \right) , \quad P _ { 0 } ( 9,9,9 ) , \quad \mathbf { u } = 3 \mathbf { i } + 4 \mathbf { j }$
A) $\frac { 26 } { 15 }$
B) $\frac { 13 } { 15 }$
C) $\frac { 52 } { 45 }$
D) $\frac { 26 } { 45 }$

Question

Quizplus · Accepted Answer

The directional derivative is found by taking the dot product of the gradient of the function with the unit vector in the direction of u. The gradient is $\nabla f = \left( \frac{2x}{x^2 + 9y^2 - 7z^2}, \frac{18y}{x^2 + 9y^2 - 7z^2}, \frac{-14z}{x^2 + 9y^2 - 7z^2} \right)$. Evaluating at $P_0$ gives $\nabla f(9,9,9)$. The unit vector $\mathbf{u}$ is $\frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j}$. The dot product gives the directional derivative, which simplifies to $\frac{26}{45}$.

Find the Derivative of the Function at P0 in the Direction