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The problem you’ve uploaded asks to find the limit:

\[
\lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}
\]

where \( f(x) = \frac{x}{\sqrt{1+x}} \).

This limit is the definition of the derivative of \( f(x) \) at \( x \). To solve it, we need to differentiate the given function \( f(x) \).

### Step-by-step solution:

#### Step 1: Find the derivative of \( f(x) \).

We have:

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

Using the quotient rule for derivatives, which states that for \( f(x) = \frac{u(x)}{v(x)} \):

\[
f'(x) = \frac{v(x) u'(x) - u(x) v'(x)}{[v(x)]^2}
\]

Here, \( u(x) = x \) and \( v(x) = \sqrt{1+x} \).

#### Step 2: Differentiate \( u(x) \) and \( v(x) \).

- \( u(x) = x \), so \( u'(x) = 1 \).
- \( v(x) = \sqrt{1+x} = (1+x)^{1/2} \), so using the chain rule:
  \[
  v'(x) = \frac{1}{2}(1+x)^{-1/2} \cdot (1) = \frac{1}{2\sqrt{1+x}}.
  \]

#### Step 3: Apply the quotient rule.

\[
f'(x) = \frac{\sqrt{1+x}(1) - x \left( \frac{1}{2\sqrt{1+x}} \right)}{(1+x)}
\]

Simplifying:

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

To simplify further, multiply the numerator by \( 2\sqrt{1+x} \) to combine the terms:

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

#### Final Result:

The derivative of \( f(x) \) is:

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

Thus, the limit as \( h \to 0 \) is:

\[
\lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} = f'(x) = \frac{2+x}{2(1+x)^{3/2}}.
\]

Let me know if you would like further details or have any questions! 

### Related Questions:
1. How would the quotient rule change if the function was \( \frac{1}{\sqrt{1+x}} \)?
2. What is the physical interpretation of the derivative in this problem?
3. How would you find the second derivative of \( f(x) \)?
4. What happens to the derivative as \( x \to \infty \)?
5. Can you verify the result using the definition of the derivative directly?

### Tip:
Always double-check your use of derivative rules (quotient, product, chain) to avoid small errors when solving calculus problems.

Find the limit: lim(h→0) (f(x+h) - f(x)) / h when f(x) = x / √(1+x)

Learn how to find the limit lim(h→0) (f(x+h) - f(x)) / h for f(x) = x / √(1+x), which involves calculating the derivative using the quotient rule. This problem explores limits, derivatives, and the quotient rule, suitable for college-level calculus students.

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