Find the derivatives of the following :
`f(x)=(x^(3)-3x^(2)+4)(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 \( f(x) = u(x) \cdot v(x) \), then the derivative \( f'(x) \) is given by:

\[ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]

### Step 1: Identify \( u \) and \( v \)

Let:
- \( u = x^3 - 3x^2 + 4 \)
- \( v = 4x^5 + x^2 - 1 \)

### Step 2: Find \( u' \) and \( v' \)

Now, we need to find the derivatives \( u' \) and \( v' \).

**Finding \( u' \):**
\[ u' = \frac{d}{dx}(x^3 - 3x^2 + 4) = 3x^2 - 6x + 0 = 3x^2 - 6x \]

**Finding \( v' \):**
\[ v' = \frac{d}{dx}(4x^5 + x^2 - 1) = 20x^4 + 2x + 0 = 20x^4 + 2x \]

### Step 3: Apply the Product Rule

Now, we can apply the product rule:
\[ f'(x) = u' \cdot v + u \cdot v' \]

Substituting \( u, v, u', \) and \( v' \):
\[ f'(x) = (3x^2 - 6x)(4x^5 + x^2 - 1) + (x^3 - 3x^2 + 4)(20x^4 + 2x) \]

### Step 4: Expand the Expression

Now we will expand both terms.

**Expanding \( (3x^2 - 6x)(4x^5 + x^2 - 1) \):**
\[ = 3x^2 \cdot 4x^5 + 3x^2 \cdot x^2 + 3x^2 \cdot (-1) - 6x \cdot 4x^5 - 6x \cdot x^2 - 6x \cdot (-1) \]
\[ = 12x^7 + 3x^4 - 3x^2 - 24x^6 - 6x^3 + 6x \]

Combining like terms:
\[ = 12x^7 - 24x^6 + 3x^4 - 6x^3 + 3x^2 + 6x \]

**Expanding \( (x^3 - 3x^2 + 4)(20x^4 + 2x) \):**
\[ = x^3 \cdot 20x^4 + x^3 \cdot 2x - 3x^2 \cdot 20x^4 - 3x^2 \cdot 2x + 4 \cdot 20x^4 + 4 \cdot 2x \]
\[ = 20x^7 + 2x^4 - 60x^6 - 6x^3 + 80x^4 + 8x \]

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

### Step 5: Combine All Terms

Now we combine both expanded results:
\[ f'(x) = (12x^7 - 24x^6 + 3x^4 - 6x^3 + 3x^2 + 6x) + (20x^7 - 60x^6 + 82x^4 - 6x^3 + 8x) \]

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

### Final Answer

Thus, the derivative of the function is:
\[ f'(x) = 32x^7 - 84x^6 + 85x^4 - 12x^3 + 14x \]