The image provided contains a mathematical problem involving partial differentiation. The task is to find the first partial derivatives of the function $g = \ln(x^2 + y^2 + z^2 + 8)$ with respect to $x$, $y$, and $z$.

Let's solve this step-by-step.

Function

$g = \ln(x^2 + y^2 + z^2 + 8)$

Partial Derivatives

1. Partial derivative with respect to $x$ ($\frac{\partial g}{\partial x}$)

Applying the chain rule: $\frac{\partial g}{\partial x} = \frac{1}{x^2 + y^2 + z^2 + 8} \cdot \frac{\partial}{\partial x}(x^2 + y^2 + z^2 + 8)$

Calculate the derivative inside: $\frac{\partial}{\partial x}(x^2 + y^2 + z^2 + 8) = 2x$

Thus: $\frac{\partial g}{\partial x} = \frac{2x}{x^2 + y^2 + z^2 + 8}$

2. Partial derivative with respect to $y$ ($\frac{\partial g}{\partial y}$)

Similarly, applying the chain rule: $\frac{\partial g}{\partial y} = \frac{1}{x^2 + y^2 + z^2 + 8} \cdot \frac{\partial}{\partial y}(x^2 + y^2 + z^2 + 8)$

Calculate the derivative inside: $\frac{\partial}{\partial y}(x^2 + y^2 + z^2 + 8) = 2y$

Thus: $\frac{\partial g}{\partial y} = \frac{2y}{x^2 + y^2 + z^2 + 8}$

3. Partial derivative with respect to $z$ ($\frac{\partial g}{\partial z}$)

Similarly, applying the chain rule: $\frac{\partial g}{\partial z} = \frac{1}{x^2 + y^2 + z^2 + 8} \cdot \frac{\partial}{\partial z}(x^2 + y^2 + z^2 + 8)$

Calculate the derivative inside: $\frac{\partial}{\partial z}(x^2 + y^2 + z^2 + 8) = 2z$

Thus: $\frac{\partial g}{\partial z} = \frac{2z}{x^2 + y^2 + z^2 + 8}$

Summary of Partial Derivatives

$\frac{\partial g}{\partial x} = \frac{2x}{x^2 + y^2 + z^2 + 8}, \quad \frac{\partial g}{\partial y} = \frac{2y}{x^2 + y^2 + z^2 + 8}, \quad \frac{\partial g}{\partial z} = \frac{2z}{x^2 + y^2 + z^2 + 8}$

Would you like more details on this process, or do you have any specific questions?

Related Questions

1. How do you apply the chain rule for multivariable functions?
2. What are the applications of partial derivatives in physics?
3. Can you explain how to find higher-order partial derivatives for the given function?
4. How do you interpret partial derivatives geometrically?
5. What happens to the partial derivatives if we change the base of the logarithm?

Tip

Remember, when computing partial derivatives, treat other variables as constants and differentiate with respect to the chosen variable.