To differentiate the function [tex]\( f(x) = (x^4 - 1)(x - 1) \)[/tex] using the product rule, let's proceed with the following steps:
1. Identify the functions to differentiate:
Let [tex]\( u(x) = x^4 - 1 \)[/tex] and [tex]\( v(x) = x - 1 \)[/tex].
2. Recall the product rule:
For two functions [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex], the product rule states that:
[tex]\[
(u \cdot v)' = u' \cdot v + u \cdot v'
\][/tex]
3. Find the derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
- [tex]\( u(x) = x^4 - 1 \)[/tex]
[tex]\[
u'(x) = \frac{d}{dx}(x^4 - 1) = 4x^3
\][/tex]
- [tex]\( v(x) = x - 1 \)[/tex]
[tex]\[
v'(x) = \frac{d}{dx}(x - 1) = 1
\][/tex]
4. Apply the product rule:
Using the product rule:
[tex]\[
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\][/tex]
Plugging in the identified functions and their derivatives:
[tex]\[
f'(x) = 4x^3 \cdot (x - 1) + (x^4 - 1) \cdot 1
\][/tex]
5. Simplify the expression:
Expand 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]
6. Combine like terms for the final result:
[tex]\[
f'(x) = x^4 + 4x^4 - 4x^3 - 1
\][/tex]
[tex]\[
f'(x) = x^4 + 4x^3 (x - 1) - 1
\][/tex]
So, the derivative of [tex]\( f(x) = (x^4 - 1)(x - 1) \)[/tex] using the product rule is:
[tex]\[
f'(x) = x^4 + 4x^3 (x - 1) - 1
\][/tex]
