To find the derivative of \( 2 \sec^2(5x) + 2x \), we will use the following rules:

1. The derivative of \( \sec^2(u) \) with respect to \( u \) is \( 2\sec^2(u) \tan(u) \).
2. The chain rule: \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \).
3. The derivative of \( 2x \) is simply \( 2 \).

Let's differentiate each term:

### Step 1: Differentiate \( 2 \sec^2(5x) \)

First, apply the chain rule. The outer function is \( \sec^2(5x) \) and the inner function is \( 5x \).

\[
\frac{d}{dx} [2 \sec^2(5x)] = 2 \cdot \left(2 \sec^2(5x) \tan(5x)\right) \cdot \frac{d}{dx}(5x)
\]

The derivative of \( 5x \) is \( 5 \), so:

\[
= 2 \cdot 2 \sec^2(5x) \tan(5x) \cdot 5 = 20 \sec^2(5x) \tan(5x)
\]

### Step 2: Differentiate \( 2x \)

The derivative of \( 2x \) is simply \( 2 \).

### Step 3: Combine both results

So the derivative of \( 2 \sec^2(5x) + 2x \) is:

\[
20 \sec^2(5x) \tan(5x) + 2
\]

### Final Answer:
\[
\frac{d}{dx} \left( 2 \sec^2(5x) + 2x \right) = 20 \sec^2(5x) \tan(5x) + 2
\]

Would you like more details or have any questions? 

Here are 5 related questions to expand your understanding:
1. How would you differentiate \( \sec(3x^2 + 1) \)?
2. What is the derivative of \( \tan(x^2) + x^3 \)?
3. How does the chain rule apply to the function \( \sin^2(4x) \)?
4. Can you find the second derivative of \( 2 \sec^2(5x) + 2x \)?
5. What are some applications of the derivative of \( \sec^2(x) \) in physics?

**Tip:** Always identify inner and outer functions clearly when using the chain rule for derivatives.

Learn how to compute the derivative of 2 sec^2(5x) + 2x step-by-step using the chain rule and derivative formulas for trigonometric functions. This solution covers advanced calculus concepts, including the chain rule and differentiation of secant functions. Suitable for Grades 11-12 or early college students.

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