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The image contains a set of calculus problems focused on finding the first derivative of various functions. Here is a breakdown of the 10 problems presented, each requiring you to compute the derivative \( f'(x) \).

### 1. \( f(x) = 2x^4 \cdot \left( \sqrt{x^5 - 6x} \right) \)
Use the product rule and chain rule. First, differentiate \( 2x^4 \), and then apply the chain rule to \( \sqrt{x^5 - 6x} \).

### 2. \( f(x) = (x^2 + 2x) \cdot (3x - 1) \)
Expand the expression and differentiate the result.

### 3. \( f(x) = (x + 5)^4 \cdot (3 - 2x) \)
Use the product rule: first differentiate \( (x + 5)^4 \), then differentiate \( (3 - 2x) \).

### 4. \( f(x) = \frac{x^2 + 5x}{2 - x} \)
Use the quotient rule: differentiate the numerator \( x^2 + 5x \) and the denominator \( 2 - x \).

### 5. \( f(x) = \frac{x - 1}{(x^2 + 3x)^2} \)
Apply the quotient rule and chain rule for the denominator.

### 6. \( f(x) = 3x^5 \cdot \sec x \)
Use the product rule: differentiate \( 3x^5 \) and \( \sec x \).

### 7. \( f(x) = 3 \cos x - \sin x \)
Differentiate directly using standard derivatives of sine and cosine functions.

### 8. \( f(x) = \sin^3(4x^2 + 5) \)
Apply the chain rule: differentiate \( \sin^3 u \) where \( u = 4x^2 + 5 \), and then differentiate \( u \).

### 9. \( f(x) = \cos 7x - \tan 3x^2 \)
Use the chain rule for both terms, starting with \( \cos 7x \) and \( \tan 3x^2 \).

### 10. \( f(x) = \sin(5x - 8) + \cos^3 7x \)
Differentiate the sine and cosine terms separately, applying the chain rule to \( \cos^3 7x \).

Let me know which specific problem you'd like me to solve, or if you'd prefer detailed solutions for all of them!

Here are some related questions to explore further:

1. How does the chain rule apply to composite functions like \( \sin^3(4x^2 + 5) \)?
2. Can you explain the steps involved in applying the product rule?
3. What are some common pitfalls when using the quotient rule in derivatives?
4. How do you simplify trigonometric functions before differentiating?
5. In what situations do you need to apply both product and chain rules in a single problem?

### Tip:
When solving problems that involve multiple differentiation rules, like product and chain rule together, break the problem down into smaller steps to avoid mistakes.

Find the first derivative of the following functions f(x): 1) f(x) = 2x^4 (√(x^5 - 6x)), 2) f(x) = (x^2 + 2x)(3x - 1), 3) f(x) = (x + 5)^4(3 - 2x), 4) f(x) = (x^2 + 5x) / (2 - x), 5) f(x) = (x - 1) / (x^2 + 3x)^2, 6) f(x) = 3x^5 sec(x), 7) f(x) = 3 cos(x) - sin(x), 8) f(x) = sin^3(4x^2 + 5), 9) f(x) = cos(7x) - tan(3x^2), 10) f(x) = sin(5x - 8) + cos^3(7x)

Solve first-order derivatives of functions involving polynomials and trigonometric expressions. This problem set applies differentiation rules such as the product rule, quotient rule, and chain rule, covering advanced calculus topics. Suitable for college-level students.

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