https://file.aiquickdraw.com/mathbot/file/1726106626059b6g8wwar.jpg

The function provided in the image is:

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

We need to differentiate this function. I'll use the quotient rule since it's a ratio of two expressions.

### Quotient Rule:
If \( f(x) = \frac{g(x)}{h(x)} \), then
\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2}
\]
Here, \( g(x) = x \) and \( h(x) = x + \frac{c}{x} \).

### Step-by-step differentiation:

1. **Differentiate the numerator**:
   \( g(x) = x \), so \( g'(x) = 1 \).

2. **Differentiate the denominator**:
   \( h(x) = x + \frac{c}{x} \), which is a sum of terms.
   - The derivative of \( x \) is \( 1 \).
   - The derivative of \( \frac{c}{x} \) is \( -\frac{c}{x^2} \) using the power rule.

   Thus, \( h'(x) = 1 - \frac{c}{x^2} \).

3. **Apply the quotient rule**:
   \[
   f'(x) = \frac{(1)(x + \frac{c}{x}) - (x)(1 - \frac{c}{x^2})}{(x + \frac{c}{x})^2}
   \]

Let me simplify this expression further.

First, expanding the terms in the numerator:

\[
1 \cdot \left( x + \frac{c}{x} \right) = x + \frac{c}{x}
\]

\[
x \cdot \left( 1 - \frac{c}{x^2} \right) = x - \frac{c}{x}
\]

Now subtract the terms:

\[
(x + \frac{c}{x}) - (x - \frac{c}{x}) = \frac{c}{x} + \frac{c}{x} = \frac{2c}{x}
\]

Thus, the derivative simplifies to:

\[
f'(x) = \frac{\frac{2c}{x}}{\left( x + \frac{c}{x} \right)^2}
\]

This is the derivative of the given function.

Would you like a more detailed breakdown or have any questions?

---

Here are 5 related questions for further exploration:
1. How would the derivative change if the constant \( c \) were negative?
2. What is the behavior of the function as \( x \to \infty \)?
3. How does the second derivative \( f''(x) \) behave?
4. Can this derivative be simplified further under certain assumptions about \( x \) or \( c \)?
5. How can this result be used to find local extrema (minima or maxima) of the function?

**Tip:** When applying the quotient rule, always simplify the expressions before finalizing the result to avoid unnecessary complexity.

Differentiate the function f(x) = x / (x + (c/x))

Learn how to differentiate the function f(x) = x / (x + (c/x)) step-by-step using the quotient rule. This calculus problem covers key concepts such as the quotient rule and basic derivatives. Suitable for students in Grades 11-12 or early college.

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