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Let's solve the given derivatives step by step:

### 1. **First Example:**
   - **Find \( F'(x) \) if \( F(x) = \sqrt{x^2 + 1} \)**.
     
     This is a composition of functions (square root and polynomial), so we use the chain rule.

     \[
     F(x) = (x^2 + 1)^{\frac{1}{2}}
     \]

     Applying the chain rule:
     \[
     F'(x) = \frac{1}{2}(x^2 + 1)^{-\frac{1}{2}} \cdot \frac{d}{dx}(x^2 + 1)
     \]
     \[
     F'(x) = \frac{1}{2}(x^2 + 1)^{-\frac{1}{2}} \cdot 2x
     \]
     \[
     F'(x) = \frac{x}{\sqrt{x^2 + 1}}
     \]

---

### 2. **Differentiate the following functions:**

   **(a) \( y = (x^3 - 1)^{100} \)**

   Again, use the chain rule here:
   \[
   \frac{dy}{dx} = 100(x^3 - 1)^{99} \cdot \frac{d}{dx}(x^3 - 1)
   \]
   \[
   \frac{dy}{dx} = 100(x^3 - 1)^{99} \cdot 3x^2
   \]
   \[
   \frac{dy}{dx} = 300x^2(x^3 - 1)^{99}
   \]

---

   **(b) \( y = e^{\sin x} \)**

   Using the chain rule again, since it's an exponential function:
   \[
   \frac{dy}{dx} = e^{\sin x} \cdot \frac{d}{dx}(\sin x)
   \]
   \[
   \frac{dy}{dx} = e^{\sin x} \cdot \cos x
   \]

---

   **(c) \( g(t) = \left( \frac{t - 1}{2t + 1} \right)^9 \)**

   This is a power of a quotient, so we need both the chain rule and the quotient rule.

   Let:
   \[
   u(t) = \frac{t - 1}{2t + 1}
   \]

   First, differentiate \( u(t) \) using the quotient rule:
   \[
   \frac{du}{dt} = \frac{(2t + 1) \cdot 1 - (t - 1) \cdot 2}{(2t + 1)^2}
   \]
   \[
   \frac{du}{dt} = \frac{(2t + 1) - 2(t - 1)}{(2t + 1)^2}
   \]
   \[
   \frac{du}{dt} = \frac{2t + 1 - 2t + 2}{(2t + 1)^2}
   \]
   \[
   \frac{du}{dt} = \frac{3}{(2t + 1)^2}
   \]

   Now apply the chain rule:
   \[
   \frac{dg}{dt} = 9 \left( \frac{t - 1}{2t + 1} \right)^8 \cdot \frac{du}{dt}
   \]
   \[
   \frac{dg}{dt} = 9 \left( \frac{t - 1}{2t + 1} \right)^8 \cdot \frac{3}{(2t + 1)^2}
   \]
   \[
   \frac{dg}{dt} = \frac{27(t - 1)^8}{(2t + 1)^{10}}
   \]

---

Would you like more details or further clarification?

Here are 5 related questions to deepen understanding:
1. How does the chain rule apply when differentiating composite functions?
2. What is the general form of the quotient rule in calculus?
3. Can you find the second derivative of \( F(x) = \sqrt{x^2 + 1} \)?
4. How do you differentiate a function raised to a power, such as \( (x^3 - 1)^{100} \)?
5. What is the derivative of \( e^{x^2} \) using the chain rule?

**Tip:** When differentiating, always break down complex expressions into simpler ones by using basic rules (like chain or product rules) step-by-step.

Differentiate the given functions: a) y = (x^3 - 1)^{100}, b) y = e^{sin(x)}, c) g(t) = (t - 1)/(2t + 1)^9. Also, find F'(x) if F(x) = sqrt(x^2 + 1).

This page provides a step-by-step solution to differentiating advanced functions like y = (x^3 - 1)^{100}, y = e^{sin(x)}, and g(t) = (t - 1)/(2t + 1)^9. Learn how to apply the chain rule and quotient rule to solve these problems. Suitable for high school or introductory college calculus students.

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