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.