Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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]
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]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.