Find the derivative of `(x^(3) - 3x^(2) + 4)(4x^(5) + x^(2) -1)` :

Options

A. `2x(10 x^(3) + 1)(x^(3) - 3x^(2) + 4) + 3x (x-2)(4x^(5) + x^(2) - 1)`
B. `2x(10 x^(3) +1)(x^(3) + 3x^(2) + 4) + 3x(x-2)(4x^(5) + x^(2)-1)`
C. `2x(10 x^(3) +1)(x^(3) - 3x^(2) + 4) + 3x(x+2)(4x^(5) + x^(2)+1)`
D. `2x(10 x^(3) +1)(x^(3) + 3x^(2) + 4) + 3x(x-2)(4x^(5) + x^(2)+1)`

Answer

To find the derivative of the function \( y = (x^3 - 3x^2 + 4)(4x^5 + x^2 - 1) \), 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:

\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step 1: Identify the functions
Let:
- \( u = x^3 - 3x^2 + 4 \)
- \( v = 4x^5 + x^2 - 1 \)

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

1. **Differentiate \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(x^3 - 3x^2 + 4) = 3x^2 - 6x \]

2. **Differentiate \( v \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(4x^5 + x^2 - 1) = 20x^4 + 2x \]

### 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 \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):

\[ \frac{dy}{dx} = (x^3 - 3x^2 + 4)(20x^4 + 2x) + (4x^5 + x^2 - 1)(3x^2 - 6x) \]

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

1. **First term**:
\[ (x^3 - 3x^2 + 4)(20x^4 + 2x) = 20x^7 - 60x^6 + 80x^4 + 2x^4 - 6x^3 + 8 \]
Combining like terms:
\[ = 20x^7 - 60x^6 + 82x^4 - 6x^3 + 8 \]

2. **Second term**:
\[ (4x^5 + x^2 - 1)(3x^2 - 6x) = 12x^7 - 24x^6 + 3x^4 - 6x^2 - 3x^2 + 6 \]
Combining like terms:
\[ = 12x^7 - 24x^6 + 3x^4 - 9x^2 + 6 \]

### Step 5: Combine both terms
Now combine both simplified terms:

\[ \frac{dy}{dx} = (20x^7 - 60x^6 + 82x^4 - 6x^3 + 8) + (12x^7 - 24x^6 + 3x^4 - 9x^2 + 6) \]

Combining like terms:

\[ = (20x^7 + 12x^7) + (-60x^6 - 24x^6) + (82x^4 + 3x^4) + (-6x^3) + (-9x^2) + (8 + 6) \]
\[ = 32x^7 - 84x^6 + 85x^4 - 6x^3 - 9x^2 + 14 \]

### Final Answer
Thus, the derivative of the given function is:

\[ \frac{dy}{dx} = 32x^7 - 84x^6 + 85x^4 - 6x^3 - 9x^2 + 14 \]