find the third derivative of 1+square root of x over square root of x

Let's solve for the third derivative of the expression:

\[
f(x) = \frac{1 + \sqrt{x}}{\sqrt{x}}
\]

### Step 1: Simplify the function
We can first simplify the expression:

\[
f(x) = \frac{1}{\sqrt{x}} + \frac{\sqrt{x}}{\sqrt{x}} = \frac{1}{\sqrt{x}} + 1
\]

Now, recall that \( \frac{1}{\sqrt{x}} = x^{-\frac{1}{2}} \), so:

\[
f(x) = x^{-\frac{1}{2}} + 1
\]

### Step 2: Find the first derivative
To find the first derivative of \( f(x) = x^{-\frac{1}{2}} + 1 \), apply the power rule:

\[
f'(x) = -\frac{1}{2} x^{-\frac{3}{2}} + 0 = -\frac{1}{2} x^{-\frac{3}{2}}
\]

### Step 3: Find the second derivative
Now, differentiate the first derivative:

\[
f''(x) = \frac{3}{4} x^{-\frac{5}{2}}
\]

### Step 4: Find the third derivative
Differentiate the second derivative:

\[
f'''(x) = -\frac{15}{8} x^{-\frac{7}{2}}
\]

Thus, the third derivative of \( f(x) = \frac{1 + \sqrt{x}}{\sqrt{x}} \) is:

\[
f'''(x) = -\frac{15}{8} x^{-\frac{7}{2}}
\]

Would you like further details or explanations on any part of the process?

Here are five related questions for further exploration:
1. What is the physical significance of higher-order derivatives in real-world problems?
2. How can the chain rule be used when differentiating more complex functions?
3. What are common errors to avoid when simplifying before differentiating?
4. How do higher-order derivatives help in curve sketching or analyzing graphs?
5. How do we approach derivatives for functions involving trigonometric or exponential terms?

**Tip:** Always simplify the function as much as possible before differentiating—it can save you time and reduce errors.

Find the third derivative of 1 + square root of x over square root of x.

Learn how to find the third derivative of the expression (1 + √x) / √x using the power rule. This calculus problem involves simplifying the function and applying higher-order differentiation techniques. Suitable for advanced high school and college-level calculus students.

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