To find the differential of \( z = \ln(x + \sqrt{y}) \), we need to compute the total derivative \( dz \). We do this by applying the chain rule for multivariable functions.

### Step 1: Differentiate \( z \) with respect to \( x \)
\[
\frac{\partial z}{\partial x} = \frac{1}{x + \sqrt{y}} \cdot \frac{\partial}{\partial x}(x + \sqrt{y}) = \frac{1}{x + \sqrt{y}}.
\]
Since \( x + \sqrt{y} \) contains \( x \), the derivative of \( x \) with respect to \( x \) is just 1, and the term \( \sqrt{y} \) is treated as constant with respect to \( x \).

### Step 2: Differentiate \( z \) with respect to \( y \)
\[
\frac{\partial z}{\partial y} = \frac{1}{x + \sqrt{y}} \cdot \frac{\partial}{\partial y}(x + \sqrt{y}) = \frac{1}{x + \sqrt{y}} \cdot \frac{1}{2\sqrt{y}}.
\]
Here, we apply the chain rule to the square root term: \( \frac{d}{dy}(\sqrt{y}) = \frac{1}{2\sqrt{y}} \).

### Step 3: Write the total differential \( dz \)
The total differential is:
\[
dz = \frac{\partial z}{\partial x} \, dx + \frac{\partial z}{\partial y} \, dy.
\]
Substituting the partial derivatives, we get:
\[
dz = \frac{1}{x + \sqrt{y}} \, dx + \frac{1}{2\sqrt{y}(x + \sqrt{y})} \, dy.
\]

Thus, the differential of \( z = \ln(x + \sqrt{y}) \) is:
\[
dz = \frac{1}{x + \sqrt{y}} \, dx + \frac{1}{2\sqrt{y}(x + \sqrt{y})} \, dy.
\]

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do we apply the chain rule in multivariable calculus?
2. What is the differential of \( z = \ln(xy) \)?
3. How do we differentiate functions involving square roots?
4. What are some practical applications of differentials in physics?
5. How do we find the second derivative of \( z = \ln(x + \sqrt{y}) \)?

**Tip:** When differentiating logarithmic functions, always check if the argument involves multiple variables so you can apply the chain rule correctly.

Discover how to find the differential of z = ln(x + √y) using multivariable calculus and the chain rule. This solution covers key steps in differentiating logarithmic functions with multiple variables, including partial derivatives and the chain rule. Ideal for university-level calculus students.

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