Hey helpful member of this forum, I am currently in a calculus and vectors self learning course, and I need help with a problem. ***As I wrote this thread I was doing all the math again. I found that my problem is my finding of the derivative of $\displaystyle y = u(u^2 + 3)^3 $. I kept the rest of my solution up just incase this is also a problem. My main concern, however, is my bold statement showing where the main problem (I think) is.***

Question:Use the chain rule to find dy/dx at the indicated value of x.

$\displaystyle y = u(u^2 + 3)^3, u = (x + 3)^2, x = -2 $

My solution:

Basically how the text explains the chain rule is: "If f and g are functions having derivatives, then the composite function $\displaystyle h(x) = f(g(x)) $ has a derivative given by $\displaystyle h'(x) = f'(g(x))g'(x) $"

So with this information this is how my solution looks:

First I expanded $\displaystyle u = (x+3)^2 $:

$\displaystyle u = x^2 + 6x + 9 $

Which makes it easier for me to find the derivative (just me likely):

$\displaystyle du/dx = (1)(2)x^{2-1} + (1)(6)x^{1-1} $

$\displaystyle = 2x + 6 $

I believe this is where the problem lies!!! So help in finding the derivative here would be awesome!!!

Then I found the derivative of $\displaystyle y = u(u^2 + 3)^3 $:

$\displaystyle dy/du = (u)(3)(u^2 + 3)^{3-1}((1)(2)u^{2-1}) $

$\displaystyle dy/du = 3u(u^2 + 3)^2(2u) $

$\displaystyle dy/du = 6u^2(u^2 + 3)^2 $*

*According to wolfram alpha, this derivative is incorrect. I do not understand why. The solution on wolfram alpha when I input $\displaystyle y = u(u^2 + 3)^2 $ is $\displaystyle (3 + u^2)^2 (3 + 7u^2) $. I do not understand how they come to this answer.* Link: http://www.wolframalpha.com/input/?i=y+%3D+u%28u^2+%2B+3%29^3

Assuming the above is correct, subbing $\displaystyle u = (x + 3)^2 $ into $\displaystyle y = u(u^2 + 3)^3 $ should *assuming the answer in the book is correct* come to 320 when I sub in -2 for x. So:

$\displaystyle (6((x)^2 + 6(x) + 9)^2)((((x)^2 + 6(x) + 9)^2) +3)^2 $

Then I sub in x = -2:

$\displaystyle (6((-2)^2 + 6(-2) + 9)^2)((((-2)^2 + 6(-2) + 9)^2) +3)^2 $

$\displaystyle = (6((1)^2))(4)^2 $

$\displaystyle = (6)(16) $

$\displaystyle = 96 $ Then multiply by 2x + 6:

$\displaystyle (96)(2) $

$\displaystyle = 192 $

When I use the derivative shown on wolfram alpha I do get 320. *I tried this before but did not get correct answer, just tried again now and did!!!* How do I find the derivative shown on wolfram alpha?

$\displaystyle (3 + (x^2 + 6x + 9)^2)^2 * (3 + 7(x^2 + 6x + 9)^2) $

Sub in x = -2:

$\displaystyle ( 3 + ((-2)^2 + 6(-2) + 9)^2)^2 * (3 + 7((-2)^2 + 6(-2) + 9)^2) $

$\displaystyle = ((3 + 1)^2)(3 + 7(1)) $

$\displaystyle = (16)(10) $

$\displaystyle = (160)(2) $ *multiply by (2x + 6) with x = -2 subbed in*

$\displaystyle = 320 $

- Thanks for looking at my thread!