 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/16/a/solution

A directional derivative measures change per unit distance, so first normalize the direction:

v = (4, 3) /5

. The gradient is

\nabla f = - \frac{2 ( x , y )}{( 1 + x ^{2} + y ^{2} ) ^{2}}, \nabla f (1, 0) = (- 1/2, 0) .

D_{v} f (1, 0) = (- 1/2, 0) \cdot (4/5, 3/5) = - \frac{2}{5} .

Using the unnormalized vector would instead give

- 2

, the change per unit parameter along

(1, 0) + t v

; it is not the rate per unit distance requested here.

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