Solve the problem.
-Find the derivative of the function $f ( x , y ) = e ^ { x y }$ at the point $( 0 , - 6 )$ in the direction in which the function decreases most rapidly.
A) -12
B) -18
C) -7
D) -6

Question

Quizplus · Accepted Answer

The gradient of the function at point $(0,-6)$ is $\nabla f(0,-6) = \langle -6,0\rangle$. This gives the direction in which the function increases most rapidly. The direction in which the function decreases most rapidly is the opposite direction, which is $\vec{v} = \langle 6,0\rangle$.
The directional derivative of the function in the direction of $\vec{v}$ is given by $\nabla f(0,-6)\cdot \frac{\vec{v}}{\|\vec{v}\|}$. Since $\|\vec{v}\| = 6$, we have
$$\nabla f(0,-6)\cdot \frac{\vec{v}}{\|\vec{v}\|} = \langle -6,0\rangle\cdot\langle 1,0\rangle = -6.$$
Therefore, the answer is \boxed{\text{(D) } -6}.

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