https://file.aiquickdraw.com/mathbot/file/1726843626781i0sg7sly.jpg

Let's solve both parts of the problem shown in the image.

### Part d)
We are given the function:
\[
f(x) = (5x - 1)^{10} \cdot (3x + 2)^2
\]
and we are asked to find the derivative using the rules of differentiation (product rule and chain rule).

To find \( f'(x) \), we will apply the product rule:

\[
(fg)' = f'g + fg'
\]

Let:
- \( u(x) = (5x - 1)^{10} \)
- \( v(x) = (3x + 2)^2 \)

#### Step 1: Derivative of \( u(x) \)
Using the chain rule for \( u(x) = (5x - 1)^{10} \):

\[
u'(x) = 10(5x - 1)^9 \cdot 5 = 50(5x - 1)^9
\]

#### Step 2: Derivative of \( v(x) \)
Using the chain rule for \( v(x) = (3x + 2)^2 \):

\[
v'(x) = 2(3x + 2)^1 \cdot 3 = 6(3x + 2)
\]

#### Step 3: Apply the product rule
Now applying the product rule to \( f(x) = u(x) \cdot v(x) \):

\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Substitute \( u'(x) \), \( u(x) \), \( v(x) \), and \( v'(x) \):

\[
f'(x) = 50(5x - 1)^9 \cdot (3x + 2)^2 + (5x - 1)^{10} \cdot 6(3x + 2)
\]

This is the derivative for part d. The expression can be simplified further if required.

---

### Part e)
We are given the function:
\[
h(x) = \frac{x}{\sqrt{2x - 5}}
\]
and we are asked to find the derivative using the quotient rule and chain rule.

#### Step 1: Apply the quotient rule
The quotient rule states:
\[
\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}
\]
Here, \( u(x) = x \) and \( v(x) = \sqrt{2x - 5} = (2x - 5)^{1/2} \).

#### Step 2: Derivatives of \( u(x) \) and \( v(x) \)
- \( u'(x) = 1 \)
- To find \( v'(x) \), we use the chain rule:
  \[
  v'(x) = \frac{1}{2}(2x - 5)^{-1/2} \cdot 2 = \frac{1}{\sqrt{2x - 5}}
  \]

#### Step 3: Apply the quotient rule
Substitute these into the quotient rule:

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

Simplifying the denominator and numerator:

\[
h'(x) = \frac{\sqrt{2x - 5} - \frac{x}{\sqrt{2x - 5}}}{2x - 5}
\]

Multiplying both terms in the numerator by \( \sqrt{2x - 5} \):

\[
h'(x) = \frac{(2x - 5) - x}{(2x - 5)\sqrt{2x - 5}} = \frac{x - 5}{(2x - 5)\sqrt{2x - 5}}
\]

Thus, this is the derivative for part e.

---

Do you want more details on any part of the solution, or help simplifying further?

Here are five related questions:
1. How does the product rule work with more than two functions?
2. What are the applications of the chain rule in physics?
3. Can you explain the steps of simplifying derivatives further?
4. What is the geometric meaning of a derivative?
5. How does the quotient rule compare with other rules of differentiation?

**Tip**: Always simplify your derivative expressions step by step to avoid errors, especially when applying multiple rules like product or chain rules.

Use the rules of differentiation to find the derivatives of f(x) = (5x - 1)^{10}(3x + 2)^2 and h(x) = x / sqrt(2x - 5).

This page explains how to find derivatives of functions involving multiple differentiation rules. We explore the product rule for f(x) = (5x - 1)^{10}(3x + 2)^2 and the quotient rule for h(x) = x / sqrt(2x - 5), providing a step-by-step solution using chain and product rules.

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