Evaluate differentiation of y with respect to x :
`y=x^(3)+(4)/(3)x^(2)-5x+1`

Explanation

To evaluate the differentiation of \( y \) with respect to \( x \), where

\[ y = x^3 + \frac{4}{3}x^2 - 5x + 1, \]

we will use the power rule of differentiation, which states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \).

### Step-by-Step Solution:

1. **Identify the function**:
We have \( y = x^3 + \frac{4}{3}x^2 - 5x + 1 \).

2. **Differentiate each term**:
- For \( x^3 \):
\[ \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2. \]
- For \( \frac{4}{3}x^2 \):
\[ \frac{d}{dx}\left(\frac{4}{3}x^2\right) = \frac{4}{3} \cdot 2x^{2-1} = \frac{8}{3}x. \]
- For \( -5x \):
\[ \frac{d}{dx}(-5x) = -5. \]
- For the constant \( 1 \):
\[ \frac{d}{dx}(1) = 0. \]

3. **Combine the derivatives**:
Now, we combine all the derivatives we calculated:
\[ \frac{dy}{dx} = 3x^2 + \frac{8}{3}x - 5. \]

4. **Final result**:
Thus, the differentiation of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 3x^2 + \frac{8}{3}x - 5. \]