Certainly! To differentiate the function [tex]\( f(x) = (x^4 - 1)(x - 1) \)[/tex] using the product rule, we follow these steps:
### Step 1: Identify the components of the product
We have two functions in the product:
[tex]\( u(x) = x^4 - 1 \)[/tex]
[tex]\( v(x) = x - 1 \)[/tex]
### Step 2: Recall the product rule
The product rule states that if you have a function [tex]\( f(x) = u(x)v(x) \)[/tex], then the derivative [tex]\( f'(x) \)[/tex] is given by:
[tex]\[ f'(x) = u'(x)v(x) + u(x)v'(x) \][/tex]
### Step 3: Differentiate each component
Differentiate [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
1. Differentiate [tex]\( u(x) \)[/tex]:
[tex]\[ u'(x) = \frac{d}{dx}(x^4 - 1) = 4x^3 \][/tex]
2. Differentiate [tex]\( v(x) \)[/tex]:
[tex]\[ v'(x) = \frac{d}{dx}(x - 1) = 1 \][/tex]
### Step 4: Apply the product rule
Now apply the product rule:
[tex]\[ f'(x) = u'(x)v(x) + u(x)v'(x) \][/tex]
Substitute [tex]\( u(x) \)[/tex], [tex]\( u'(x) \)[/tex], [tex]\( v(x) \)[/tex], and [tex]\( v'(x) \)[/tex] into the product rule equation:
[tex]\[ f'(x) = (4x^3)(x - 1) + (x^4 - 1)(1) \][/tex]
### Step 5: Simplify the expression
Distribute and combine like terms:
[tex]\[ f'(x) = 4x^3(x - 1) + (x^4 - 1) \][/tex]
[tex]\[ f'(x) = 4x^3 \cdot x - 4x^3 + x^4 - 1 \][/tex]
[tex]\[ f'(x) = 4x^4 - 4x^3 + x^4 - 1 \][/tex]
Combine the [tex]\( x^4 \)[/tex] terms:
[tex]\[ f'(x) = (4x^4 + x^4) - 4x^3 - 1 \][/tex]
[tex]\[ f'(x) = 5x^4 - 4x^3 - 1 \][/tex]
So the derivative of the function [tex]\( f(x) = (x^4 - 1)(x - 1) \)[/tex] is:
[tex]\[ f'(x) = 5x^4 - 4x^3 - 1 \][/tex]
And that's the step-by-step solution for differentiating the given function!
