Question

Express partial derivative u divided by partial derivative x ​, partial derivative u divided by partial derivative y ​, and partial derivative u divided by partial derivative z as functions of​ x, y, and z both by using the chain rule and by expressing u directly in terms of​ x, y, and z before differentiating.

Answer 1

To find the expressions for the partial derivatives of u with respect to x, y, and z, we can use the chain rule or express u directly in terms of x, y, and z before differentiating.

Step-by-step explanation:

To find the expressions for the partial derivatives of u with respect to x, y, and z, we can use the chain rule. Let's start with the partial derivative of u with respect to x (du/dx). We can express this as:

du/dx = (du/dx)(dx/dx) + (du/dy)(dy/dx) + (du/dz)(dz/dx)

Since dx/dx = 1, dy/dx = 0, and dz/dx = 0, the equation simplifies to:

du/dx = (du/dx)

We can follow a similar process to find the expressions for du/dy and du/dz. By expressing u directly in terms of x, y, and z before differentiating, we can avoid using the chain rule. For example, if u = f(x,y,z), then:

du/dx = df/dx

du/dy = df/dy

du/dz = df/dz