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It looks like you're working on a Calculus I assignment that involves applying the **Product and Quotient Rules** for derivatives. Let's break down the questions in the image and provide some help.

### Problem (a)
You are asked to find the derivative of:

\[
f(x) = (x^2 - 3x - 1)(5x^2 + 1)
\]

You can apply the **Product Rule** here, which is:

\[
\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)
\]

1. Compute \( f'(x) \) (derivative of \( x^2 - 3x - 1 \)):
   \[
   f'(x) = 2x - 3
   \]

2. Compute \( g'(x) \) (derivative of \( 5x^2 + 1 \)):
   \[
   g'(x) = 10x
   \]

Then, apply the Product Rule formula:

\[
f'(x) = (2x - 3)(5x^2 + 1) + (x^2 - 3x - 1)(10x)
\]

You can expand and simplify this expression if needed.

---

### Problem (b)
You are asked to find the derivative of:

\[
g(x) = (x + 4\sqrt{x})(7x - 1)
\]

This is also a **Product Rule** problem.

1. First, simplify \( g(x) = (x + 4x^{1/2})(7x - 1) \).
2. Compute the derivative of \( x + 4x^{1/2} \):
   \[
   \frac{d}{dx} \left( x + 4x^{1/2} \right) = 1 + 2x^{-1/2}
   \]
3. Compute the derivative of \( 7x - 1 \):
   \[
   \frac{d}{dx} (7x - 1) = 7
   \]

Apply the Product Rule:

\[
g'(x) = (1 + 2x^{-1/2})(7x - 1) + (x + 4x^{1/2})(7)
\]

This can also be expanded and simplified if necessary.

---

### Problem (c)
You are asked to differentiate a quotient:

\[
h(x) = \frac{3x + 1}{5x + 2}
\]

This requires the **Quotient Rule**, which is given as:

\[
\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}
\]

Here, \( f(x) = 3x + 1 \) and \( g(x) = 5x + 2 \).

1. Compute \( f'(x) = 3 \).
2. Compute \( g'(x) = 5 \).

Now, apply the Quotient Rule:

\[
h'(x) = \frac{(5x + 2)(3) - (3x + 1)(5)}{(5x + 2)^2}
\]

Simplify the numerator and leave the denominator as \( (5x + 2)^2 \).

---

Would you like a detailed step-by-step simplification for any of these problems?

### 5 Related Questions:
1. How do you apply the Chain Rule when differentiating more complex functions?
2. Can the Product Rule be extended to more than two functions?
3. How can you combine the Quotient Rule with the Chain Rule for more advanced derivatives?
4. What are the common errors to avoid when applying the Product and Quotient Rules?
5. How do you check your work when simplifying derivative expressions?

### Tip:
Always simplify your derivatives fully, as this will make future steps, such as finding critical points, much easier!

Use the product and quotient rules to find the derivative of the functions.

This calculus problem covers how to find the derivative of functions using both the product rule and quotient rule. Step-by-step solutions provided for functions such as f(x) = (x^2 - 3x - 1)(5x^2 + 1) and g(x) = (x + 4√x)(7x - 1). Suitable for Calculus I students.

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