The expression you've provided seems to be the derivative of the function:

$\frac{d}{dx} \left[ 2 \sec^2(x) \tan(x) + \sin(x) \right]$

Let’s compute this derivative step by step.

1. Differentiate $2 \sec^2(x) \tan(x)$

We'll use the product rule for differentiation. The product rule states that if we have two functions $u(x)$ and $v(x)$, then:

$\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

Here, $u(x) = 2 \sec^2(x)$ and $v(x) = \tan(x)$.

• The derivative of $u(x) = 2 \sec^2(x)$ is:

  $u'(x) = 2 \cdot 2 \sec^2(x) \tan(x) = 4 \sec^2(x) \tan(x)$

• The derivative of $v(x) = \tan(x)$ is:

  $v'(x) = \sec^2(x)$

Using the product rule:

$\frac{d}{dx} \left[ 2 \sec^2(x) \tan(x) \right] = 4 \sec^2(x) \tan^2(x) + 2 \sec^4(x)$

2. Differentiate $\sin(x)$

The derivative of $\sin(x)$ is:

$\frac{d}{dx} \left[ \sin(x) \right] = \cos(x)$

3. Combine the results

Now, putting everything together:

$\frac{d}{dx} \left[ 2 \sec^2(x) \tan(x) + \sin(x) \right] = 4 \sec^2(x) \tan^2(x) + 2 \sec^4(x) + \cos(x)$

Thus, the derivative is:

$\boxed{4 \sec^2(x) \tan^2(x) + 2 \sec^4(x) + \cos(x)}$

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Would you like any further details on the steps or have any questions? Here are some related questions you might find interesting:

1. What is the derivative of $\sec(x)$ and how does it relate to $\tan(x)$?
2. How can you apply the product rule in different types of trigonometric expressions?
3. What are the rules for differentiating trigonometric functions like $\tan(x)$ and $\sec(x)$?
4. How do you simplify expressions with multiple trigonometric functions after differentiation?
5. What are some common mistakes when applying the product rule to trigonometric functions?

Tip: When using the product rule for trigonometric functions, always double-check the derivatives of each function and simplify terms carefully.