https://r2.erweima.ai/mathbot/image/1726266778210d5cyrziq.jpg

The image contains a mathematical problem asking to compute the limit:

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

This is the definition of the derivative of a function \( f(x) \) at a point \( x \). The problem asks to compute this limit for three different functions:

a) \( f(x) = 2x + 3 \)  
b) \( f(x) = x^3 + x + 1 \)  
c) \( f(x) = 5 - 2x - 3x^2 \)

Let's proceed by calculating the derivative for each of these functions.

### a) \( f(x) = 2x + 3 \)

For this linear function, the derivative is straightforward:
\[
f'(x) = \lim_{h \to 0} \frac{(2(x+h) + 3) - (2x + 3)}{h}
= \lim_{h \to 0} \frac{2x + 2h + 3 - 2x - 3}{h}
= \lim_{h \to 0} \frac{2h}{h} = 2
\]
Thus, \( f'(x) = 2 \).

### b) \( f(x) = x^3 + x + 1 \)

For this cubic function, let's apply the definition of the derivative:
\[
f'(x) = \lim_{h \to 0} \frac{((x+h)^3 + (x+h) + 1) - (x^3 + x + 1)}{h}
\]
First expand \( (x+h)^3 \):
\[
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
\]
Now substitute this into the difference quotient:
\[
f'(x) = \lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3 + x + h + 1) - (x^3 + x + 1)}{h}
\]
Simplify the expression:
\[
f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 + h}{h}
\]
Factor out \( h \):
\[
f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2 + 1)
\]
As \( h \to 0 \), this becomes:
\[
f'(x) = 3x^2 + 1
\]

### c) \( f(x) = 5 - 2x - 3x^2 \)

For this quadratic function:
\[
f'(x) = \lim_{h \to 0} \frac{(5 - 2(x+h) - 3(x+h)^2) - (5 - 2x - 3x^2)}{h}
\]
First expand \( (x+h)^2 \):
\[
(x+h)^2 = x^2 + 2xh + h^2
\]
Substitute into the difference quotient:
\[
f'(x) = \lim_{h \to 0} \frac{5 - 2x - 2h - 3(x^2 + 2xh + h^2) - (5 - 2x - 3x^2)}{h}
\]
Simplify:
\[
f'(x) = \lim_{h \to 0} \frac{-2h - 3(2xh + h^2)}{h} = \lim_{h \to 0} \frac{-2h - 6xh - 3h^2}{h}
\]
Factor out \( h \):
\[
f'(x) = \lim_{h \to 0} (-2 - 6x - 3h)
\]
As \( h \to 0 \), this becomes:
\[
f'(x) = -2 - 6x
\]

### Summary of Derivatives:
- a) \( f'(x) = 2 \)
- b) \( f'(x) = 3x^2 + 1 \)
- c) \( f'(x) = -2 - 6x \)

Would you like further details or explanations on any of these steps?

---

Here are 5 related questions:
1. How does the derivative represent the slope of a function at a point?
2. What are the rules for differentiating polynomial functions?
3. How would the process change if \( f(x) \) were a trigonometric function?
4. Can we apply the product rule or chain rule to these functions, and how?
5. What does it mean when the derivative of a function is zero?

**Tip**: The derivative of a constant is always zero, since a constant function has no change in its slope.

Encuentre el límite \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \) para cada función: \( a) f(x) = 2x + 3 \), \( b) f(x) = x^3 + x + 1 \), \( c) f(x) = 5 - 2x - 3x^2 \).

Learn how to calculate derivatives of polynomial functions using the limit definition of the derivative. Functions include linear, cubic, and quadratic examples such as \( f(x) = 2x + 3 \), \( f(x) = x^3 + x + 1 \), and \( f(x) = 5 - 2x - 3x^2 \). Suitable for high school and undergraduate students.

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