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

1 Answer
Sep 26, 2015

Answer:

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)#