Question

How do you find the derivative of `(2x+1)^5 (x^3-x+1)^4`?

Answer 1

Use the product, power and chain rules.

Explanation:

`f(x) = (2x+1)^5 (x^3-x+1)^4`

Use the product, power and chain rules. I use the product rule in the order, `(uv)' = u'v+uv'`

`f'(x) = [5(2x+1)^4 d/dx(2x+1)] (x^3-x+1)^4 color(red)(+)(2x+1)^5 [4(x^3-x+1)^3 d/dx(x^3-x+1)]`

`= [5(2x+1)^4 (2)] (x^3-x+1)^4color(red)(+)(2x+1)^5 [4(x^3-x+1)^3 (3x^2-1)]`

We have finished differentiating, but we can "clean up" the expression. First rewrite each of the two terms (with the red `color(red)(+)` between them.

`= 10(2x+1)^4 (x^3-x+1)^4color(red)(+)4(2x+1)^5 (x^3-x+1)^3 (3x^2-1)`

Now remove the common facotrs of the two terms.

`= 2(2x+1)^4 (x^3-x+1)^3[5(x^3-x+1) color(red)(+) 2(2x+1)(3x^2-1)]`

Finally simplify what was left behind in the brackets.

`= 2(2x+1)^4 (x^3-x+1)^3[5x^3-5x+5color(red)(+)12x^3+6x^2-4x-2]`

`= 2(2x+1)^4 (x^3-x+1)^3(17x^3+6x^2-9x+3)`