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el tema de matematicas es derivada de orden superior y derivada implicita


To solve the problem in the image, we need to find the derivative of the given function using the rules of differentiation.

### Function provided:
\[
h(x) = 4x^2 - 7x - 5
\]

### Step-by-step derivative calculation:
The derivative of a function is found by applying the power rule. The power rule states that for any term of the form \( ax^n \), the derivative is \( a \cdot n \cdot x^{n-1} \).

1. **For \( 4x^2 \):**
   \[
   \frac{d}{dx}(4x^2) = 2 \cdot 4 \cdot x^{2-1} = 8x
   \]

2. **For \( -7x \):**
   \[
   \frac{d}{dx}(-7x) = -7
   \]

3. **For \( -5 \):** Since the derivative of a constant is 0:
   \[
   \frac{d}{dx}(-5) = 0
   \]

### Putting it all together:
\[
h'(x) = 8x - 7
\]

Thus, the correct answer is:
\[
h'(x) = 8x - 7
\]

The correct option is **the third one**.

---

Would you like more details on any step? Here are 5 related questions for further practice:

1. How would you calculate the second derivative of \( h(x) \)?
2. What is the derivative of \( h(x) = x^3 - 4x^2 + x - 1 \)?
3. How does the chain rule work in differentiation?
4. How do you apply implicit differentiation?
5. How do you find the derivative of a product of two functions?

**Tip:** Always simplify each term separately when using the power rule for derivatives!

Calcule la derivada de la siguiente función: h(x) = 4x^2 - 7x - 5

Learn how to compute the derivative of the function h(x) = 4x^2 - 7x - 5 using the power rule. Step-by-step explanation and solution suitable for high school students (Grades 10-12).

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