GATE 2024 AG – Question 48
Find the directional derivative of $u=x^2y+y^2z$ at(1,2,3) in direction $\hat i+2\hat j+3\hat k$,to two decimals.
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Show answer and explanation
Correct answer: 11.20 to 11.30
Explanation
Gradient is $\nabla u=(2xy,x^2+2yz,y^2)$,which becomes(4,13,4). Normalize the direction to $(1,2,3)/\sqrt{14}$. The derivative is the dot product $(4+26+12)/\sqrt{14}=42/\sqrt{14}=11.22497$,rounded **11.22**. Omitting normalization would give42 instead of the rate per unit distance.