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 \( x \) for the function

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

we will follow the rules of differentiation step by step.

### Step 1: Identify the function
The function we need to differentiate is

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

### Step 2: Differentiate each term
We will differentiate each term of the function using the power rule of differentiation, which states that if \( y = x^n \), then

\[ \frac{dy}{dx} = nx^{n-1}. \]

1. Differentiate \( x^3 \):
\[ \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2. \]

2. Differentiate \( \frac{4}{3}x^2 \):
\[ \frac{d}{dx}\left(\frac{4}{3}x^2\right) = \frac{4}{3} \cdot 2x^{2-1} = \frac{8}{3}x. \]

3. Differentiate \( -5x \):
\[ \frac{d}{dx}(-5x) = -5. \]

4. Differentiate the constant \( 1 \):
\[ \frac{d}{dx}(1) = 0. \]

### Step 3: Combine the results
Now, we combine the results of the differentiation:

\[ \frac{dy}{dx} = 3x^2 + \frac{8}{3}x - 5 + 0. \]

Thus, we can simplify this to:

\[ \frac{dy}{dx} = 3x^2 + \frac{8}{3}x - 5. \]

### Final Answer
The differentiation of \( y \) with respect to \( x \) is

\[ \frac{dy}{dx} = 3x^2 + \frac{8}{3}x - 5. \]

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