How do you find the first partial derivatives of the function u = 2xyz?

To find the first partial derivatives of the function u = 2xyz, we will differentiate the function with respect to each variable while treating the other variables as constants.

1. Partial Derivative with respect to x

To find the partial derivative of u with respect to x, we differentiate u = 2xyz while treating y and z as constants:

∂u/∂x = 2yz

2. Partial Derivative with respect to y

Next, we find the partial derivative of u with respect to y. This time, we treat x and z as constants:

∂u/∂y = 2xz

3. Partial Derivative with respect to z

Finally, we calculate the partial derivative of u with respect to z, treating x and y as constants:

∂u/∂z = 2xy

Conclusion

In summary, the first partial derivatives of the function u = 2xyz are:

• ∂u/∂x = 2yz
• ∂u/∂y = 2xz
• ∂u/∂z = 2xy

These derivatives can be useful in various applications such as optimization and analyzing the behavior of the function in multivariable calculus.