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GATE 2024 EE – Question 36

Engineering Mathematics · Calculus: Directional derivatives · 2 marks · Multiple choice

Consider a vector $\bar u=2\hat x+\hat y+2\hat z$, where $\hat x,\hat y,\hat z$ represent unit vectors along the coordinate axes $x,y,z$ respectively. The directional derivative of the function $f(x,y,z)=2\ln(xy)+\ln(yz)+3\ln(xz)$ at the point $(x,y,z)=(1,1,1)$ in the direction of $\bar u$ is

  1. 0
  2. $\dfrac7{5\sqrt2}$
  3. 7
  4. 21

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Correct answer: (C) 7

Explanation

$\nabla f=\left(\dfrac2x+\dfrac3x,\ \dfrac2y+\dfrac1y,\ \dfrac1z+\dfrac3z\right)=(5,3,4)$ at $(1,1,1)$. The unit vector along $\bar u$ is $\dfrac13(2,1,2)$, so the directional derivative is $\dfrac{10+3+8}{3}=7$.