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Differentiate the following using the product rule:

[tex]\[ f(x) = \left(x^4 - 1\right)(x - 1) \][/tex]


Sagot :

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]