https://r2.erweima.ai/mathbot/image/1726196713491iukak7bi.jpg

Let's solve the problem step by step:

We are given the function:
\[
h(x) = 4x^2 - 7x - 5
\]

The goal is to find the derivative \( h'(x) \) using basic differentiation rules.

### Step 1: Differentiate each term

- The derivative of \( 4x^2 \) is \( 8x \), using the power rule \( \frac{d}{dx}(x^n) = n x^{n-1} \).
- The derivative of \( -7x \) is \( -7 \), because the derivative of a linear term \( ax \) is just the coefficient \( a \).
- The derivative of the constant \( -5 \) is 0, since the derivative of any constant is 0.

### Step 2: Combine the results

So, the derivative of \( h(x) \) is:
\[
h'(x) = 8x - 7
\]

### Conclusion:
The correct answer is:
\[
h'(x) = 8x - 7
\]

This corresponds to the third option in the image.

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Do you want any more details on the derivative calculation, or have any questions?

Here are 5 additional related questions to explore further:
1. What is the derivative of \( x^3 - 5x^2 + 3x - 8 \)?
2. How would you calculate the second derivative of \( h(x) = 4x^2 - 7x - 5 \)?
3. What is the geometric interpretation of the derivative of a function?
4. How does the power rule work for derivatives with fractional exponents?
5. What is the derivative of trigonometric functions like \( \sin(x) \) or \( \cos(x) \)?

**Tip:** Always check each term individually when differentiating complex functions to avoid mistakes in applying the rules.

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

Learn how to calculate the derivative of the quadratic function h(x) = 4x^2 - 7x - 5 using the power rule. This detailed solution is suitable for Grade 11-12 or college-level students studying calculus.

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