To differentiate the function [tex]\( f(x) = (x^4 - 1)(x - 1) \)[/tex] using the product rule, we will follow these steps systematically:
1. Identify the functions: Let [tex]\( u(x) = x^4 - 1 \)[/tex] and [tex]\( v(x) = x - 1 \)[/tex].
2. Differentiate each function:
- The derivative of [tex]\( u(x) \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( u'(x) = 4x^3 \)[/tex].
- The derivative of [tex]\( v(x) \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( v'(x) = 1 \)[/tex].
3. Apply the product rule: The product rule states that if [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]
4. Plug in the differentiated functions and the original functions:
- Substitute [tex]\( u'(x) = 4x^3 \)[/tex], [tex]\( u(x) = x^4 - 1 \)[/tex], [tex]\( v'(x) = 1 \)[/tex], and [tex]\( v(x) = x - 1 \)[/tex] into the product rule formula:
[tex]\[
f'(x) = 4x^3 (x - 1) + (x^4 - 1)(1)
\][/tex]
5. Simplify the expression:
[tex]\[
f'(x) = 4x^3 (x - 1) + x^4 - 1
\][/tex]
[tex]\[
f'(x) = 4x^4 - 4x^3 + x^4 - 1
\][/tex]
6. Combine like terms:
[tex]\[
f'(x) = x^4 + 4x^3 (x - 1) - 1
\][/tex]
Therefore, the derivative of [tex]\( f(x) = (x^4 - 1)(x - 1) \)[/tex] with respect to [tex]\( x \)[/tex] is
[tex]\[
f'(x) = x^4 + 4x^3 (x - 1) - 1
\][/tex]
