Find the directional derivative of f=xyz at (1,1,1) in the direction of 𝑖⃗ +𝑗⃗ +𝑘⃗

To find the directional derivative of f=xyz at (1,1,1) in the direction of 𝑖⃗ +𝑗⃗ +𝑘⃗, we can use the formula:

Df = ∇f · 𝑢

where ∇f is the gradient of f and 𝑢 is the unit vector in the direction of 𝑖⃗ +𝑗⃗ +𝑘⃗.

Step 1: Calculate the gradient of f The gradient of f is given by:

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

To find the partial derivatives, we differentiate f with respect to each variable separately:

∂f/∂x = yz ∂f/∂y = xz ∂f/∂z = xy

So, the gradient of f is:

∇f = (yz, xz, xy)

Step 2: Calculate the unit vector 𝑢 The unit vector 𝑢 in the direction of 𝑖⃗ +𝑗⃗ +𝑘⃗ is given by:

𝑢 = (𝑖⃗ +𝑗⃗ +𝑘⃗) / ||𝑖⃗ +𝑗⃗ +𝑘⃗||

To calculate ||𝑖⃗ +𝑗⃗ +𝑘⃗||, we find the magnitude of the vector:

||𝑖⃗ +𝑗⃗ +𝑘⃗|| = √(1^2 + 1^2 + 1^2) = √3

So, the unit vector 𝑢 is:

𝑢 = (𝑖⃗ +𝑗⃗ +𝑘⃗) / √3

Step 3: Calculate the directional derivative Now, we can substitute the values into the formula:

Df = ∇f · 𝑢

Df = (yz, xz, xy) · (𝑖⃗ +𝑗⃗ +𝑘⃗) / √3

Df = (yz/√3) + (xz/√3) + (xy/√3)

Therefore, the directional derivative of f=xyz at (1,1,1) in the direction of 𝑖⃗ +𝑗⃗ +𝑘⃗ is (yz/√3) + (xz/√3) + (xy/√3).