How do you differentiate # f(x)=(2x+1)(4-x^2)(1+x^2) # using the product rule?

Question

Isaiah Allen · Accepted Answer

#10x^4+4x^3-18x^2-6x-8#

You could further simply it if you are required to do so.

Multiply #(1+x^2)# by #(4-x^2)#: #(4+3x^2-x^4)# #((2x+1)'(4+3x^2-x^4))+((2x+1)(4+3x^2-x^4)')# #(2(4+3x^2-x^4))+((2x+1)(6x-4x^3))# #(8+6x^2-2x^4)+(6x+12x^2-4x^3-8x^4)# #8+6x^2-2x^4+6x+12x^2-4x^3-8x^4# #8+6x+18x^2-4x^3-10x^4# Optional, divide by #-1#, #10x^4+4x^3-18x^2-6x-8# Good Luck

Samantha Adkison · Answer

undefined To differentiate $ f(x)=(2x+1)(4-x^2)(1+x^2) $ using the product rule, follow these steps:

1. Identify the functions $ u $ and $ v $:
   $ u = 2x + 1 $
   $ v = (4 - x^2)(1 + x^2) $

2. Apply the product rule:
   $ f'(x) = u'v + uv' $

3. Differentiate $ u $ and $ v $:
   $ u' = 2 $
   $ v' = (4 - x^2)'(1 + x^2) + (4 - x^2)(1 + x^2)' $

4. Simplify $ u' $ and $ v' $:
   $ (4 - x^2)' = -2x $
   $ (1 + x^2)' = 2x $

5. Substitute $ u' $ and $ v' $ into the product rule:
   $ f'(x) = 2(4 - x^2)(1 + x^2) + (2x + 1)(-2x) + (4 - x^2)(1 + x^2)(2x) $

6. Expand and simplify the expression:
   $ f'(x) = 2(4 - x^2)(1 + x^2) - 2x(2x + 1) + 2x(4 - x^2)(1 + x^2) $

7. Combine like terms if necessary.