Question

Let \( a \) be the magnitude of the directional derivative of the function: \[\phi(x, y) = \frac{x}{x^2 + y^2}\] along a line making an angle of \( 45^\circ \) with the positive x-axis at the point \( (0, 2) \). Then, the value of \( 1/a^2 \) is:

• A. 24
• B. \( \frac{1}{4\sqrt2} \)
• C. \( 16\sqrt2 \)
• D. 32

Hint:

The directional derivative gives the rate of change of a function in the direction of a vector, and it is computed using the dot product of the gradient and the unit vector in the direction of interest.

Answer 1

The Correct Option is D

We first compute the gradient of \( \phi(x, y) \) and then find the directional derivative along the given direction. The gradient is: \[ \nabla \phi = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y} \right) \] After calculating the gradient, we compute the magnitude of the directional derivative and then find \( 1/a^2 \).