` y = log [4^(2x) ((x^(2)+5)/(sqrt(2x^(3)-4)))^(3//2)], ` find ` (dy)/(dx)`

Answer

To find \(\frac{dy}{dx}\) for the function

\[ y = \log \left[ 4^{2x} \left( \frac{x^2 + 5}{\sqrt{2x^3 - 4}} \right)^{\frac{3}{2}} \right], \]

we will follow these steps:

### Step 1: Simplify the logarithmic expression using properties of logarithms.

Using the property \(\log(ab) = \log a + \log b\) and \(\log(a^b) = b \log a\), we can rewrite \(y\):

\[ y = \log \left( 4^{2x} \right) + \log \left( \left( \frac{x^2 + 5}{\sqrt{2x^3 - 4}} \right)^{\frac{3}{2}} \right). \]

This simplifies to:

\[ y = 2x \log 4 + \frac{3}{2} \log \left( \frac{x^2 + 5}{\sqrt{2x^3 - 4}} \right). \]

### Step 2: Further simplify the logarithmic expression.

Using the property \(\log \left( \frac{a}{b} \right) = \log a - \log b\):

\[ y = 2x \log 4 + \frac{3}{2} \left( \log(x^2 + 5) - \log(\sqrt{2x^3 - 4}) \right). \]

Since \(\log(\sqrt{a}) = \frac{1}{2} \log a\), we can rewrite it as:

\[ y = 2x \log 4 + \frac{3}{2} \log(x^2 + 5) - \frac{3}{4} \log(2x^3 - 4). \]

### Step 3: Differentiate \(y\) with respect to \(x\).

Now, we differentiate \(y\):

\[ \frac{dy}{dx} = \frac{d}{dx} \left( 2x \log 4 \right) + \frac{3}{2} \frac{d}{dx} \left( \log(x^2 + 5) \right) - \frac{3}{4} \frac{d}{dx} \left( \log(2x^3 - 4) \right). \]

Calculating each derivative:

1. \(\frac{d}{dx} (2x \log 4) = 2 \log 4\) (since \(\log 4\) is a constant).
2. \(\frac{d}{dx} \left( \log(x^2 + 5) \right) = \frac{1}{x^2 + 5} \cdot \frac{d}{dx}(x^2 + 5) = \frac{2x}{x^2 + 5}\).
3. \(\frac{d}{dx} \left( \log(2x^3 - 4) \right) = \frac{1}{2x^3 - 4} \cdot \frac{d}{dx}(2x^3 - 4) = \frac{6x^2}{2x^3 - 4}\).

Putting it all together:

\[ \frac{dy}{dx} = 2 \log 4 + \frac{3}{2} \cdot \frac{2x}{x^2 + 5} - \frac{3}{4} \cdot \frac{6x^2}{2x^3 - 4}. \]

### Step 4: Simplify the expression.

This gives us:

\[ \frac{dy}{dx} = 2 \log 4 + \frac{3x}{x^2 + 5} - \frac{9x^2}{2(2x^3 - 4)}. \]

### Final Answer:

Thus, the derivative \(\frac{dy}{dx}\) is:

\[ \frac{dy}{dx} = 2 \log 4 + \frac{3x}{x^2 + 5} - \frac{9x^2}{2(2x^3 - 4)}. \]