Can you simplify in steps the calculating the final expression for dy/dx using the above problem in step 3


Sure! Let's break down the simplification of the final expression for \( \frac{dy}{dx} \) step by step from where we applied the quotient rule.

Recall, we have:

\[
\frac{dy}{dx} = \frac{(3x^2) \cdot \left( \frac{1}{2\sqrt{x}} \cdot (2x - 4) + 2\sqrt{x} \right) - \left( \sqrt{x} \cdot (2x - 4) \right) \cdot 6x}{(3x^2)^2}
\]

### Step 1: Simplify the numerator term by term.

The numerator consists of two terms:

- First term: \( (3x^2) \cdot \left( \frac{1}{2\sqrt{x}} \cdot (2x - 4) + 2\sqrt{x} \right) \)
- Second term: \( - \left( \sqrt{x} \cdot (2x - 4) \right) \cdot 6x \)

#### Step 1a: Expand the first term.
We first expand the inner expression:

\[
\frac{1}{2\sqrt{x}} \cdot (2x - 4) = \frac{2x - 4}{2\sqrt{x}} = \frac{x - 2}{\sqrt{x}}
\]
Thus, the expression becomes:
\[
(3x^2) \cdot \left( \frac{x - 2}{\sqrt{x}} + 2\sqrt{x} \right)
\]

Now, break it into two parts:
\[
(3x^2) \cdot \frac{x - 2}{\sqrt{x}} + (3x^2) \cdot 2\sqrt{x}
\]

These simplify as:
- First part: \( 3x^2 \cdot \frac{x - 2}{\sqrt{x}} = 3x^2 \cdot \left( x^{1/2} - 2x^{-1/2} \right) = 3x^{5/2} - 6x^{3/2} \)
- Second part: \( 3x^2 \cdot 2\sqrt{x} = 6x^{5/2} \)

Thus, the expanded first term is:
\[
3x^{5/2} - 6x^{3/2} + 6x^{5/2}
\]
Combining like terms:
\[
9x^{5/2} - 6x^{3/2}
\]

#### Step 1b: Expand the second term.
Now, expand the second term:
\[
- \left( \sqrt{x} \cdot (2x - 4) \right) \cdot 6x = -6x \cdot \left( \sqrt{x} \cdot (2x - 4) \right)
\]
Expanding inside:
\[
\sqrt{x} \cdot (2x - 4) = 2x^{3/2} - 4x^{1/2}
\]
So the expression becomes:
\[
-6x \cdot (2x^{3/2} - 4x^{1/2}) = -12x^{5/2} + 24x^{3/2}
\]

### Step 2: Combine the two terms of the numerator.
Now, combine the results from both expansions:

\[
\left( 9x^{5/2} - 6x^{3/2} \right) + \left( -12x^{5/2} + 24x^{3/2} \right)
\]
This simplifies to:
\[
(9x^{5/2} - 12x^{5/2}) + (-6x^{3/2} + 24x^{3/2}) = -3x^{5/2} + 18x^{3/2}
\]

### Step 3: Simplify the denominator.
The denominator is:
\[
(3x^2)^2 = 9x^4
\]

### Step 4: Final simplification.
Thus, we have:
\[
\frac{dy}{dx} = \frac{-3x^{5/2} + 18x^{3/2}}{9x^4}
\]

Now, simplify the powers of \( x \). Break the numerator into two parts:
\[
\frac{-3x^{5/2}}{9x^4} + \frac{18x^{3/2}}{9x^4}
\]
Simplifying each term:
\[
\frac{-3}{9} \cdot \frac{x^{5/2}}{x^4} + \frac{18}{9} \cdot \frac{x^{3/2}}{x^4} = -\frac{1}{3} \cdot x^{-3/2} + 2 \cdot x^{-5/2}
\]

Thus, the final simplified expression is:
\[
\frac{dy}{dx} = \frac{6 - x}{3x^{5/2}}
\]

This matches the result obtained earlier!

Do you have any further questions, or would you like more clarification on any specific step?

### Related Questions:
1. Why is it important to combine like terms in the simplification process?
2. How does understanding fractional exponents help with differentiation?
3. How can we determine whether the function is increasing or decreasing using the derivative?
4. What do the different parts of the quotient rule represent in practical problems?
5. How can we apply these rules to more complex rational functions?

**Tip:** Always check if terms in the numerator or denominator can be factored before simplifying—it might save you a lot of work!

Learn how to simplify the differentiation of a function using the quotient rule and fractional exponents. Step-by-step explanation of calculating dy/dx from a given problem. Suitable for advanced high school and college students.

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