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The directional derivative of function φ = xy + yz + zx at point P(3, -3, -3) in the directiontoward point Q(4, -1, -1) is
- a)-3
- b)1
- c)-2
- d)0
Correct answer is option 'C'. Can you explain this answer?
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The directional derivative of function φ = xy + yz +zx at point P...
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The directional derivative of function φ = xy + yz +zx at point P...
Solution:
Given function f(x,y,z) = xy + yz + zx and the two points P(3,-3,-3) and Q(4,-1,-1).
We need to find the directional derivative of f at P in the direction of Q.
Directional derivative of f at P in the direction of Q is given by the dot product of the gradient of f at P and the unit vector in the direction of Q.
Let's find the gradient of f at P.
Gradient of f at P = (∂f/∂x, ∂f/∂y, ∂f/∂z) at (3,-3,-3)
∂f/∂x = y + z = (-3) + (-3) = -6
∂f/∂y = x + z = 3 + (-3) = 0
∂f/∂z = y + x = (-3) + 3 = 0
So, gradient of f at P = (-6, 0, 0)
Now, let's find the unit vector in the direction of Q.
Unit vector in the direction of Q = (Q - P)/|Q - P|
= (4 - 3, -1 - (-3), -1 - (-3))/sqrt((4 - 3)^2 + (-1 - (-3))^2 + (-1 - (-3))^2)
= (1/3, 2/3, 2/3)
Therefore, directional derivative of f at P in the direction of Q = gradient of f at P . unit vector in the direction of Q
= (-6, 0, 0) . (1/3, 2/3, 2/3)
= -2
Hence, the correct option is (C) -2.
Given function f(x,y,z) = xy + yz + zx and the two points P(3,-3,-3) and Q(4,-1,-1).
We need to find the directional derivative of f at P in the direction of Q.
Directional derivative of f at P in the direction of Q is given by the dot product of the gradient of f at P and the unit vector in the direction of Q.
Let's find the gradient of f at P.
Gradient of f at P = (∂f/∂x, ∂f/∂y, ∂f/∂z) at (3,-3,-3)
∂f/∂x = y + z = (-3) + (-3) = -6
∂f/∂y = x + z = 3 + (-3) = 0
∂f/∂z = y + x = (-3) + 3 = 0
So, gradient of f at P = (-6, 0, 0)
Now, let's find the unit vector in the direction of Q.
Unit vector in the direction of Q = (Q - P)/|Q - P|
= (4 - 3, -1 - (-3), -1 - (-3))/sqrt((4 - 3)^2 + (-1 - (-3))^2 + (-1 - (-3))^2)
= (1/3, 2/3, 2/3)
Therefore, directional derivative of f at P in the direction of Q = gradient of f at P . unit vector in the direction of Q
= (-6, 0, 0) . (1/3, 2/3, 2/3)
= -2
Hence, the correct option is (C) -2.
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The directional derivative of function φ = xy + yz +zx at point P(3, -3, -3) in the directiontoward point Q(4, -1, -1) isa)-3b)1c)-2d)0Correct answer is option 'C'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2025 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about The directional derivative of function φ = xy + yz +zx at point P(3, -3, -3) in the directiontoward point Q(4, -1, -1) isa)-3b)1c)-2d)0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The directional derivative of function φ = xy + yz +zx at point P(3, -3, -3) in the directiontoward point Q(4, -1, -1) isa)-3b)1c)-2d)0Correct answer is option 'C'. Can you explain this answer?.
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