Find the derivatives of the following at any point of their domains :
`G(x)=(5x^(3)+3x-1)(x-1)`

Answer

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

\[ (uv)' = u'v + uv' \]

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

### Step 2: Differentiate \( u(x) \) and \( v(x) \)
Now we will find the derivatives of \( u(x) \) and \( v(x) \).

1. Differentiate \( u(x) \):
\[ u'(x) = \frac{d}{dx}(5x^3 + 3x - 1) = 15x^2 + 3 \]

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

### Step 3: Apply the product rule
Now we can apply the product rule:
\[ G'(x) = u'(x)v(x) + u(x)v'(x) \]

Substituting the values we found:
\[ G'(x) = (15x^2 + 3)(x - 1) + (5x^3 + 3x - 1)(1) \]

### Step 4: Expand the expression
Now we will expand both terms:
1. Expand \( (15x^2 + 3)(x - 1) \):
\[ = 15x^2 \cdot x - 15x^2 \cdot 1 + 3 \cdot x - 3 \cdot 1 = 15x^3 - 15x^2 + 3x - 3 \]

2. The second term is simply:
\[ (5x^3 + 3x - 1) \]

### Step 5: Combine like terms
Now combine the results from the expansions:
\[ G'(x) = (15x^3 - 15x^2 + 3x - 3) + (5x^3 + 3x - 1) \]
\[ = (15x^3 + 5x^3) + (-15x^2) + (3x + 3x) + (-3 - 1) \]
\[ = 20x^3 - 15x^2 + 6x - 4 \]

### Final Answer
Thus, the derivative of the function \( G(x) \) is:
\[ G'(x) = 20x^3 - 15x^2 + 6x - 4 \]