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Derivada de orden superior. 
Derivación implícita.

You have provided an image where you are asked to find the second derivative, \( f''(x) \), given the function:

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

To solve this, we need to apply the rules of differentiation, starting with the first derivative and then the second derivative.

### Step 1: First Derivative \( f'(x) \)
The function is a quotient, so we use the **quotient rule**:
\[
f'(x) = \frac{(v \cdot u') - (u \cdot v')}{v^2}
\]
where \( u(x) = 2x - 1 \) and \( v(x) = x + 1 \).

- \( u'(x) = 2 \)
- \( v'(x) = 1 \)

Now applying the quotient rule:
\[
f'(x) = \frac{(x+1)(2) - (2x-1)(1)}{(x+1)^2}
\]
Simplifying the numerator:
\[
f'(x) = \frac{2(x+1) - (2x-1)}{(x+1)^2} = \frac{2x + 2 - 2x + 1}{(x+1)^2} = \frac{3}{(x+1)^2}
\]

Thus, the first derivative is:
\[
f'(x) = \frac{3}{(x+1)^2}
\]

### Step 2: Second Derivative \( f''(x) \)
Now we differentiate \( f'(x) = \frac{3}{(x+1)^2} \).

This is of the form \( g(x) = \frac{C}{h(x)} \), where \( C = 3 \) and \( h(x) = (x+1)^2 \). Using the **quotient rule** again or simply applying the chain rule:

\[
f''(x) = -3 \cdot \frac{d}{dx} \left( (x+1)^{-2} \right)
\]
Using the power rule:
\[
f''(x) = -3 \cdot (-2) \cdot (x+1)^{-3} = \frac{6}{(x+1)^3}
\]

### Final Answer:
The second derivative of \( f(x) = \frac{2x - 1}{x + 1} \) is:
\[
f''(x) = \frac{6}{(x+1)^3}
\]

Let me know if you need more details or have any questions! Here are five related questions you might explore:

1. How would the second derivative change if the numerator of \( f(x) \) was \( 3x - 2 \)?
2. What is the physical interpretation of a second derivative in terms of motion?
3. How do you use implicit differentiation for functions involving multiple variables?
4. How does the quotient rule compare with the product rule in terms of their application?
5. Can you find higher-order derivatives, such as the third derivative, for this function?

**Tip**: When solving derivative problems involving quotients, it's often helpful to rewrite the expression in a more convenient form if possible to simplify the differentiation process.

This problem involves finding the second derivative of the function f(x) = (2x-1)/(x+1) using the quotient rule and power rule. The solution is broken down into two steps: first finding the first derivative, then using the result to compute the second derivative. This tutorial is suitable for students in Grade 11-12 studying pre-calculus or calculus.

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