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The given function is:

\[
y = \sqrt{\frac{(3x + 4)^3}{(2x - 3)^2}}
\]

To find \(\frac{dy}{dx}\), we will use a combination of chain rule and quotient rule, considering the square root.

### Step 1: Rewrite the equation in a more convenient form

The square root can be rewritten as a power of \( \frac{1}{2} \):

\[
y = \left( \frac{(3x + 4)^3}{(2x - 3)^2} \right)^{\frac{1}{2}}
\]

### Step 2: Use the chain rule

Differentiate both sides with respect to \(x\). First, apply the chain rule to handle the exponent \(\frac{1}{2}\):

\[
\frac{dy}{dx} = \frac{1}{2} \left( \frac{(3x + 4)^3}{(2x - 3)^2} \right)^{-\frac{1}{2}} \cdot \frac{d}{dx} \left( \frac{(3x + 4)^3}{(2x - 3)^2} \right)
\]

Now, we need to find the derivative of the expression inside the square root (i.e., \(\frac{(3x + 4)^3}{(2x - 3)^2}\)).

### Step 3: Use the quotient rule

To differentiate \(\frac{(3x + 4)^3}{(2x - 3)^2}\), use the quotient rule:

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

Here, \( f(x) = (3x + 4)^3 \) and \( g(x) = (2x - 3)^2 \).

#### Step 3.1: Differentiate \(f(x)\)

\[
f(x) = (3x + 4)^3
\]

Using the chain rule:

\[
f'(x) = 3 \cdot (3x + 4)^2 \cdot 3 = 9(3x + 4)^2
\]

#### Step 3.2: Differentiate \(g(x)\)

\[
g(x) = (2x - 3)^2
\]

Using the chain rule:

\[
g'(x) = 2 \cdot (2x - 3) \cdot 2 = 4(2x - 3)
\]

#### Step 3.3: Apply the quotient rule

Now, apply the quotient rule:

\[
\frac{d}{dx} \left( \frac{(3x + 4)^3}{(2x - 3)^2} \right) = \frac{9(3x + 4)^2 \cdot (2x - 3)^2 - (3x + 4)^3 \cdot 4(2x - 3)}{(2x - 3)^4}
\]

### Step 4: Substitute back into the derivative

Now substitute the result from Step 3 into the expression for \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{1}{2} \left( \frac{(3x + 4)^3}{(2x - 3)^2} \right)^{-\frac{1}{2}} \cdot \frac{9(3x + 4)^2 \cdot (2x - 3)^2 - (3x + 4)^3 \cdot 4(2x - 3)}{(2x - 3)^4}
\]

This is the derivative \(\frac{dy}{dx}\).

Would you like a further simplification or more details on any of the steps? Let me know if you have any questions!

Here are five related questions:
1. How would the result change if the exponent in the numerator were 2 instead of 3?
2. What happens to \(\frac{dy}{dx}\) if \(x = \frac{3}{2}\)?
3. Can you use logarithmic differentiation for this problem?
4. How does the quotient rule work when the denominator is a constant?
5. How would you solve this using implicit differentiation?

**Tip:** When using the chain rule, always break down the process into smaller functions for easier differentiation.

Find dy/dx for the function y = sqrt((3x + 4)^3 / (2x - 3)^2)

Learn how to differentiate the function y = sqrt((3x + 4)^3 / (2x - 3)^2) using the Chain Rule and Quotient Rule. This step-by-step solution explains key calculus concepts and formulas, suitable for Calculus I students.

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