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Let's solve each of the given functions step by step:

### 1. \( f(x) = x^5 - 2x^3 + x \)
This is a polynomial function. You can analyze it by finding the derivative, roots, and critical points if needed. The derivative is:

\[
f'(x) = 5x^4 - 6x^2 + 1
\]

### 2. \( g(s) = 2s^4 - 4s^3 + 7s - 1 \)
This is another polynomial function. The first derivative to find the rate of change is:

\[
g'(s) = 8s^3 - 12s^2 + 7
\]

### 3. \( F(x) = x^2 \sqrt{x} - 5x \)
This function combines a power and a square root. Simplify \( \sqrt{x} \) as \( x^{1/2} \) and then expand:

\[
F(x) = x^2 \cdot x^{1/2} - 5x = x^{5/2} - 5x
\]

The derivative is:

\[
F'(x) = \frac{5}{2}x^{3/2} - 5
\]

### 4. \( f(x) = \sqrt{x^2 + 1} \)
This is a composite function involving a square root. The derivative using the chain rule is:

\[
f'(x) = \frac{d}{dx} \left(\sqrt{x^2 + 1}\right) = \frac{1}{2\sqrt{x^2 + 1}} \cdot 2x = \frac{x}{\sqrt{x^2 + 1}}
\]

### 5. \( f(t) = 4 \cos(t^2) \)
This is a trigonometric function with a quadratic argument. The derivative using the chain rule is:

\[
f'(t) = 4 \cdot (-\sin(t^2)) \cdot \frac{d}{dt}(t^2) = -8t \sin(t^2)
\]

### 6. \( G(x) = \cot(t^2) \)
This function is a cotangent with a quadratic argument. The derivative using the chain rule and the derivative of cotangent is:

\[
G'(x) = -\csc^2(t^2) \cdot \frac{d}{dt}(t^2) = -2t \csc^2(t^2)
\]

Would you like a detailed explanation for any specific problem or further calculations?

Here are 5 related questions:
1. How do you find the critical points for \( f(x) = x^5 - 2x^3 + x \)?
2. What is the significance of the derivative in understanding the behavior of \( g(s) = 2s^4 - 4s^3 + 7s - 1 \)?
3. How can you apply the chain rule to more complex functions like \( F(x) = x^2 \sqrt{x} - 5x \)?
4. What are the domain restrictions for \( f(x) = \sqrt{x^2 + 1} \)?
5. How can trigonometric identities help simplify the differentiation of \( G(x) = \cot(t^2) \)?

**Tip:** When dealing with composite functions, remember to apply the chain rule correctly by differentiating the outer function first and then multiplying by the derivative of the inner function.

Explore detailed solutions for finding derivatives of polynomial, composite, and trigonometric functions. Includes step-by-step explanations and applications of key formulas such as the chain rule. Ideal for advanced high school and college students.

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