To find the partial derivatives of the function $f(x, y, z) = \sqrt{x^2 + y^2 + z^2}$ with respect to $x$, $y$, and $z$, we apply the rules of differentiation.

1. Partial derivative with respect to $x$: $\frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + y^2 + z^2}}$

2. Partial derivative with respect to $y$: $\frac{\partial f}{\partial y} = \frac{y}{\sqrt{x^2 + y^2 + z^2}}$

3. Partial derivative with respect to $z$: $\frac{\partial f}{\partial z} = \frac{z}{\sqrt{x^2 + y^2 + z^2}}$

Summary of Partial Derivatives:

• $\frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + y^2 + z^2}}$
• $\frac{\partial f}{\partial y} = \frac{y}{\sqrt{x^2 + y^2 + z^2}}$
• $\frac{\partial f}{\partial z} = \frac{z}{\sqrt{x^2 + y^2 + z^2}}$

Do you want details on any specific step or have any questions? Here are some related questions for further exploration:

1. How do the partial derivatives relate to the geometric interpretation of the function?
2. Can you explain how to apply these partial derivatives in optimization problems?
3. What are the implications of the partial derivatives being zero at certain points?
4. How would you find the second partial derivatives of this function?
5. Can you provide a physical context where this function and its derivatives might be relevant?

Tip: When calculating partial derivatives, remember that you treat all other variables as constants while differentiating with respect to one variable.