GATE 2024 EE Q.36: solution and answer

GATE 2024 · EEQ.36 · 2 marks · MCQ

Consider a vector $\overset{u}{ˉ} = 2 \overset{x}{^} + \overset{y}{^} + 2 \overset{z}{^}$ , where $\overset{x}{^}, \overset{y}{^}, \overset{z}{^}$ represent unit vectors along the coordinate axes $x, y, z$ respectively. The directional derivative of the function $f (x, y, z) = 2 ln (x y) + ln (y z) + 3 ln (x z)$ at the point $(x, y, z) = (1, 1, 1)$ in the direction of $\overset{u}{ˉ}$ is

(A)
$0$
(B)
$\frac{7}{5 2}$
(C)
$7$
(D)
$21$

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Answer and solution

ANSWER

The directional derivative of $f$ at $(1, 1, 1)$ in the direction of $\overset{u}{ˉ}$ is $7$ , which is option (C).

Official answer: CSource: Official answer key, GATE 2024 EE (IISc Bengaluru)

Split each logarithm using $ln (ab) = ln a + ln b$ . This is valid near $(1, 1, 1)$ , where $x, y, z > 0$ : $f = 2 ln x + 2 ln y + ln y + ln z + 3 ln x + 3 ln z = 5 ln x + 3 ln y + 4 ln z$
Gradient: $\nabla f = (\frac{5}{x}, \frac{3}{y}, \frac{4}{z})$ . At $(1, 1, 1)$ this is $\nabla f = 5 \overset{x}{^} + 3 \overset{y}{^} + 4 \overset{z}{^}$ .
Unit vector along $\overset{u}{ˉ}$ : $∣ \overset{u}{ˉ} ∣ = 2^{2} + 1^{2} + 2^{2} = 9 = 3$ , so $\overset{u}{^} = \frac{1}{3} (2 \overset{x}{^} + \overset{y}{^} + 2 \overset{z}{^})$ .
The directional derivative is $D_{\overset{u}{^}} f = \nabla f \cdot \overset{u}{^}$ : $D_{\overset{u}{^}} f = \frac{( 5 ) ( 2 ) + ( 3 ) ( 1 ) + ( 4 ) ( 2 )}{3} = \frac{10 + 3 + 8}{3} = \frac{21}{3} = 7$
The directional derivative is $7$ , which is option (C).

KEY IDEA

The directional derivative is the gradient dotted with the UNIT vector in the given direction: $D_{\overset{u}{^}} f = \nabla f \cdot \frac{u ˉ}{∣ u ˉ ∣}$ . Splitting the logarithms first turns $f$ into $5 ln x + 3 ln y + 4 ln z$ , and its gradient can then be read off directly.

WATCH OUT FOR

Forgetting to normalise $\overset{u}{ˉ}$ and using $\nabla f \cdot \overset{u}{ˉ} = 21$ gives option (D), which is exactly three times the correct value because $∣ \overset{u}{ˉ} ∣ = 3$ . A second trap is collecting the log coefficients wrongly. For example, $y$ appears in both $2 ln (x y)$ and $ln (y z)$ , so its coefficient is $2 + 1 = 3$ , not $2$ or $1$ .

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GATE 2024 EE Q.36: solution and answer

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