https://r2.erweima.ai/mathbot/image/1724003705411qitdo3oe.jpg

The function provided in the image is:

\[
f(x) = \left[\frac{\sqrt{x^4 - x} \cdot \sec(x^2 + 2x)}{5}\right]^5
\]

To solve this, we need to find the derivative \( f'(x) \). We'll apply the chain rule, product rule, and the derivative of trigonometric functions.

### Step 1: Simplify the function

Let \( u(x) = \frac{\sqrt{x^4 - x} \cdot \sec(x^2 + 2x)}{5} \), so the function becomes:
\[
f(x) = [u(x)]^5
\]

### Step 2: Apply the chain rule

The derivative of \( f(x) = [u(x)]^5 \) is:
\[
f'(x) = 5[u(x)]^4 \cdot u'(x)
\]

### Step 3: Find \( u'(x) \)

Now, we need to differentiate \( u(x) \):

\[
u(x) = \frac{\sqrt{x^4 - x} \cdot \sec(x^2 + 2x)}{5}
\]

We apply the product rule here. Let \( v(x) = \sqrt{x^4 - x} \) and \( w(x) = \sec(x^2 + 2x) \).

\[
u'(x) = \frac{1}{5} \left[ v'(x) \cdot w(x) + v(x) \cdot w'(x) \right]
\]

### Step 4: Differentiate \( v(x) = \sqrt{x^4 - x} \)

First, express \( v(x) \) as:
\[
v(x) = (x^4 - x)^{1/2}
\]

Differentiate using the chain rule:
\[
v'(x) = \frac{1}{2}(x^4 - x)^{-1/2} \cdot (4x^3 - 1)
\]

### Step 5: Differentiate \( w(x) = \sec(x^2 + 2x) \)

Using the chain rule:
\[
w'(x) = \sec(x^2 + 2x) \cdot \tan(x^2 + 2x) \cdot (2x + 2)
\]

### Step 6: Combine everything

Now, combine the results from Step 4 and Step 5 into the expression for \( u'(x) \):
\[
u'(x) = \frac{1}{5} \left[ \frac{4x^3 - 1}{2\sqrt{x^4 - x}} \cdot \sec(x^2 + 2x) + \sqrt{x^4 - x} \cdot \sec(x^2 + 2x) \cdot \tan(x^2 + 2x) \cdot (2x + 2) \right]
\]

### Step 7: Substitute \( u'(x) \) back into \( f'(x) \)

Finally:
\[
f'(x) = 5 \cdot \left[\frac{\sqrt{x^4 - x} \cdot \sec(x^2 + 2x)}{5}\right]^4 \cdot u'(x)
\]

This expression represents the derivative of the given function.

Would you like further details or have any questions?

### Related Questions:
1. How would you compute the derivative if the exponent in the function was different?
2. Can you apply this approach to find the second derivative of \( f(x) \)?
3. How does the complexity of the function affect the application of the chain rule?
4. What would the derivative look like if the secant function was replaced with cosine?
5. Can you explore how the derivative changes if the argument inside the secant function is altered?

**Tip:** When dealing with complex functions, breaking them down into simpler parts and applying rules like the chain rule and product rule step by step is essential for accuracy.

Learn how to calculate the derivative of a complex function step-by-step. This problem involves differentiation using the chain rule, product rule, and trigonometric functions. Suitable for college-level students studying calculus.

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