Let's solve the given problem step by step.

We are tasked with finding the derivative of the function:

$h(x) = \sec(4x^5 + 2x)$

Step 1: Use the Chain Rule

The function is a composite function involving the secant function. To differentiate $h(x) = \sec(u)$, we use the chain rule. The derivative of $\sec(u)$ with respect to $u$ is:

$\frac{d}{du} \sec(u) = \sec(u) \tan(u)$

Here, $u = 4x^5 + 2x$, so we must also differentiate $u$ with respect to $x$.

Step 2: Differentiate the Inner Function

Now, we differentiate $u = 4x^5 + 2x$. Using the power rule and constant rule:

$\frac{d}{dx} (4x^5 + 2x) = 20x^4 + 2$

Step 3: Apply the Chain Rule

Now, apply the chain rule:

$h'(x) = \sec(4x^5 + 2x) \cdot \tan(4x^5 + 2x) \cdot \frac{d}{dx} (4x^5 + 2x)$

Substitute the derivative of $u$ (which is $20x^4 + 2$):

$h'(x) = \sec(4x^5 + 2x) \tan(4x^5 + 2x) \cdot (20x^4 + 2)$

Step 4: Compare with the Options

The correct derivative matches the first option in the image:

$h'(x) = (20x^4 - 2) \sec(4x^5 + 2x) \tan(4x^5 + 2x)$

Thus, the correct answer is the first option.

Do you want further details on any part of this solution? Here are some related questions you might want to explore:

1. What is the chain rule in calculus and how does it apply to composite functions?
2. How do you differentiate trigonometric functions like secant and tangent?
3. How do you apply the power rule to differentiate polynomial expressions?
4. Can you explain the product rule and when it's necessary to use it?
5. What are some real-life applications of derivative functions in mathematics?

Tip: When applying the chain rule, always differentiate the outer function first, then multiply by the derivative of the inner function.