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 \( f(x) = (x^3 - 3x^2 + 4)(4x^5 + x^2 - 1) \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]

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

### Step 2: Differentiate \( u(x) \)
Now we differentiate \( u(x) \):
\[ u'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(3x^2) + \frac{d}{dx}(4) \]
Calculating each term:
- \( \frac{d}{dx}(x^3) = 3x^2 \)
- \( \frac{d}{dx}(3x^2) = 6x \)
- \( \frac{d}{dx}(4) = 0 \)

So,
\[ u'(x) = 3x^2 - 6x \]

### Step 3: Differentiate \( v(x) \)
Next, we differentiate \( v(x) \):
\[ v'(x) = \frac{d}{dx}(4x^5) + \frac{d}{dx}(x^2) - \frac{d}{dx}(1) \]
Calculating each term:
- \( \frac{d}{dx}(4x^5) = 20x^4 \)
- \( \frac{d}{dx}(x^2) = 2x \)
- \( \frac{d}{dx}(1) = 0 \)

So,
\[ v'(x) = 20x^4 + 2x \]

### Step 4: Apply the product rule
Now we apply the product rule:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]
Substituting the derivatives and the original functions:
\[ f'(x) = (3x^2 - 6x)(4x^5 + x^2 - 1) + (x^3 - 3x^2 + 4)(20x^4 + 2x) \]

### Step 5: Simplify the expression
Now we can simplify the expression:
1. Expand \( (3x^2 - 6x)(4x^5 + x^2 - 1) \):
- \( 3x^2 \cdot 4x^5 = 12x^7 \)
- \( 3x^2 \cdot x^2 = 3x^4 \)
- \( 3x^2 \cdot (-1) = -3x^2 \)
- \( -6x \cdot 4x^5 = -24x^6 \)
- \( -6x \cdot x^2 = -6x^3 \)
- \( -6x \cdot (-1) = 6x \)

Combining these gives:
\[ 12x^7 - 24x^6 + 3x^4 - 6x^3 - 3x^2 + 6x \]

2. Expand \( (x^3 - 3x^2 + 4)(20x^4 + 2x) \):
- \( x^3 \cdot 20x^4 = 20x^7 \)
- \( x^3 \cdot 2x = 2x^4 \)
- \( -3x^2 \cdot 20x^4 = -60x^6 \)
- \( -3x^2 \cdot 2x = -6x^3 \)
- \( 4 \cdot 20x^4 = 80x^4 \)
- \( 4 \cdot 2x = 8x \)

Combining these gives:
\[ 20x^7 - 60x^6 + (2x^4 + 80x^4) - 6x^3 + 8x = 20x^7 - 60x^6 + 82x^4 - 6x^3 + 8x \]

### Step 6: Combine all terms
Now combine all the terms from both expansions:
\[ f'(x) = (12x^7 + 20x^7) + (-24x^6 - 60x^6) + (3x^4 + 2x^4 + 80x^4) + (-6x^3 - 6x^3) + (6x + 8x) \]
This simplifies to:
\[ f'(x) = 32x^7 - 84x^6 + 85x^4 - 12x + 14 \]

### Final Result
Thus, the derivative of the function \( f(x) \) is:
\[ f'(x) = 32x^7 - 84x^6 + 85x^4 - 12x + 14 \]