Find the directional derivative of u = xyz at (1,2, 3) in the direction from (1, 2, 3) to(1,-1, -3). is....(correct upto two decimal places)Correct answer is '-3.13'. Can you explain this answer?

Question

Answer

Directional Derivative of u = xyz

To find the directional derivative of the function u = xyz at the point (1,2,3) in the direction from (1,2,3) to (1,-1,-3), we can use the formula for directional derivative:

D_vu = ∇u · v

where ∇u is the gradient of u and v is the direction vector.

Step 1: Calculate the Gradient ∇u

The gradient of u is given by:

∇u = (∂u/∂x, ∂u/∂y, ∂u/∂z)

Taking the partial derivatives of u with respect to x, y, and z, we have:

∂u/∂x = yz
∂u/∂y = xz
∂u/∂z = xy

Thus, the gradient of u is:

∇u = (yz, xz, xy)

Step 2: Calculate the Direction Vector v

The direction vector v is given by the difference between the two points:

v = (1, -1, -3) - (1, 2, 3)
= (0, -3, -6)

Step 3: Calculate the Dot Product ∇u · v

To find the directional derivative, we need to calculate the dot product of the gradient and the direction vector:

∇u · v = (yz, xz, xy) · (0, -3, -6)
= 0 + (-3xz) + (-6xy)
= -3xz - 6xy

Step 4: Evaluate the Directional Derivative at the Given Point

Finally, we substitute the values x=1, y=2, z=3 into the expression -3xz - 6xy:

D_vu = -3xz - 6xy
= -3(1)(3) - 6(1)(2)
= -9 - 12
= -21

Now, we need to round the answer to two decimal places. Rounding -21 to two decimal places gives -21.00, which is not the correct answer of -3.13.

Therefore, there may have been an error in the calculation or in the given correct answer. Please double-check the problem and the provided solution.

Answer

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