If `y = (3-x^2)(x^3-x+1)` then find`dy/dx`

Explanation

To find the derivative of the function \( y = (3 - x^2)(x^3 - x + 1) \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:

\[ \frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step 1: Identify the functions
Let:
- \( u = 3 - x^2 \)
- \( v = x^3 - x + 1 \)

### Step 2: Differentiate \( u \) and \( v \)
Now we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

1. **Differentiate \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(3 - x^2) = 0 - 2x = -2x \]

2. **Differentiate \( v \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(x^3 - x + 1) = 3x^2 - 1 + 0 = 3x^2 - 1 \]

### Step 3: Apply the product rule
Now we can apply the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

Substituting the values we found:
\[ \frac{dy}{dx} = (3 - x^2)(3x^2 - 1) + (x^3 - x + 1)(-2x) \]

### Step 4: Simplify the expression
Now we will simplify the expression:

1. **Expand the first term**:
\[ (3 - x^2)(3x^2 - 1) = 9x^2 - 3 - 3x^4 + x^2 = -3x^4 + 10x^2 - 3 \]

2. **Expand the second term**:
\[ (x^3 - x + 1)(-2x) = -2x^4 + 2x^2 - 2x \]

3. **Combine both terms**:
\[ \frac{dy}{dx} = (-3x^4 + 10x^2 - 3) + (-2x^4 + 2x^2 - 2x) \]
\[ = -5x^4 + 12x^2 - 2x - 3 \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -5x^4 + 12x^2 - 2x - 3 \]