Help with function u of x (chain rule)?

Question

Given f(x) ={(x^2 +3x +1)^5} / {(x+3)^5}, identify a function of u of x and an integer n not equal to 1 such that f(x) = u^n. Then compute f'(x).

The function is a fraction. I hope I typed it correclty in this format.

Answer 1

f(x) ={ (x^2 +3x +1)^5 } / { (x+3)^5 }

First, rewrite the equation with one power symbol:

f(x) ={ (x^2 +3x +1) / (x+3) }^5

u = (x^2 +3x +1) / (x+3)
and
f(x) = u^5

To find f'(x):

f'(x) = 5 * u^4 * (du/dx)

Then, find du/dx by using the quotient rule:

u = (x^2 +3x +1) / (x+3)
du/dx = {(x + 3)(2x + 3) - (x^2 + 3x + 1)(1) } / (x + 3)^2
={(2x^2 + 9x + 9) - (x^2 + 3x + 1) } / (x + 3)^2
={ (x^2 + 6x + 8) / (x + 3)^2 }

So, f'(x) = 5 * u^4 * { (x^2 + 6x + 8) / (x + 3)^2 }

Just replace "u" with (x^2 +3x +1) / (x+3):

f'(x) = 5 * {(x^2 +3x +1) / (x+3)}^4 * { (x^2 + 6x + 8) / (x + 3)^2 }
= { 5 * (x^2 + 6x + 8) * (x^2 +3x +1)^4 } / { (x + 3)^6 }

Answer 2

u= (x^2+3x+1)/(x+3) , n=5

f'(x) = 5[(x^2+3x+1)/(x+3)]^4
*[(x+3)(2x+3)-(x^2+3x+1)]/
(x+3)^2

I had to continue the product to the next line to make it all fit.

You're using the quotient rule to get the deriv. of your u function.