Find the derivative of the following w.r.t. x :
`y=4/3x^(3//4)`

Answer

To find the derivative of the function \( y = \frac{4}{3} x^{\frac{3}{4}} \) with respect to \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Function**:
The given function is:
\[ y = \frac{4}{3} x^{\frac{3}{4}} \]

2. **Apply the Constant Multiple Rule**:
The derivative of a constant multiplied by a function can be expressed as:
\[ \frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)] \]
Here, \( c = \frac{4}{3} \) and \( f(x) = x^{\frac{3}{4}} \).

3. **Differentiate Using the Power Rule**:
The power rule states that:
\[ \frac{d}{dx}[x^n] = n \cdot x^{n-1} \]
Applying this to \( f(x) = x^{\frac{3}{4}} \):
\[ \frac{d}{dx}[x^{\frac{3}{4}}] = \frac{3}{4} x^{\frac{3}{4} - 1} = \frac{3}{4} x^{\frac{3}{4} - \frac{4}{4}} = \frac{3}{4} x^{-\frac{1}{4}} \]

4. **Combine the Results**:
Now, substituting back into the derivative:
\[ \frac{dy}{dx} = \frac{4}{3} \cdot \frac{3}{4} x^{-\frac{1}{4}} \]

5. **Simplify**:
The \( \frac{4}{3} \) and \( \frac{3}{4} \) will cancel out:
\[ \frac{dy}{dx} = x^{-\frac{1}{4}} \]

6. **Rewrite in Fraction Form**:
We can express \( x^{-\frac{1}{4}} \) as:
\[ \frac{dy}{dx} = \frac{1}{x^{\frac{1}{4}}} \]

### Final Answer:
Thus, the derivative of \( y = \frac{4}{3} x^{\frac{3}{4}} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{x^{\frac{1}{4}}} \]