(ii) Differentiate `y=x^(3) sqrt( 3x -4)`

Answer

To differentiate the function \( y = x^3 \sqrt{3x - 4} \), we will apply the product rule of differentiation. The product rule states that if \( y = u \cdot v \), then

\[ \frac{dy}{dx} = u'v + uv' \]

where \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \) respectively.

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x^3 \)
- \( v = \sqrt{3x - 4} \)

### Step 2: Differentiate \( u \)
Now, we differentiate \( u \):
\[ u' = \frac{d}{dx}(x^3) = 3x^2 \]

### Step 3: Differentiate \( v \)
Next, we differentiate \( v \). Since \( v = (3x - 4)^{1/2} \), we will use the chain rule:
\[ v' = \frac{1}{2}(3x - 4)^{-1/2} \cdot \frac{d}{dx}(3x - 4) \]
Calculating the derivative of \( 3x - 4 \):
\[ \frac{d}{dx}(3x - 4) = 3 \]
Thus,
\[ v' = \frac{1}{2}(3x - 4)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3x - 4}} \]

### Step 4: Apply the Product Rule
Now we can apply the product rule:
\[ \frac{dy}{dx} = u'v + uv' \]
Substituting the values we found:
\[ \frac{dy}{dx} = (3x^2)(\sqrt{3x - 4}) + (x^3)\left(\frac{3}{2\sqrt{3x - 4}}\right) \]

### Step 5: Simplify the Expression
Now we simplify the expression:
\[ \frac{dy}{dx} = 3x^2\sqrt{3x - 4} + \frac{3x^3}{2\sqrt{3x - 4}} \]

To combine these terms, we can find a common denominator:
\[ \frac{dy}{dx} = \frac{6x^2(3x - 4) + 3x^3}{2\sqrt{3x - 4}} \]
Simplifying the numerator:
\[ 6x^2(3x - 4) + 3x^3 = 18x^3 - 24x^2 + 3x^3 = 21x^3 - 24x^2 \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{21x^3 - 24x^2}{2\sqrt{3x - 4}} \]

### Final Result
The derivative of \( y = x^3 \sqrt{3x - 4} \) is:
\[ \frac{dy}{dx} = \frac{21x^3 - 24x^2}{2\sqrt{3x - 4}} \]