Math Help - Please verify my working for quotient and chain rule differentiation

1. Please verify my working for quotient and chain rule differentiation

Hi everyone,

I've been trying to solve a quite complex equation using both the chain rule and the quotient rule of differentiation. I think I have a handle on the first part of the equation (the top row), but please see the attached document with my full working and verify that I am correct. Secondly, I am unsure how to proceed with solving the equation after that, since I have to manipulate the (bottom squared) part and I am unsure whether I need to expand using chain rule, or simply expand. So any help solving this problem will be much appreciated!

Here is the attached full workings:

As you can see, I now need to work with the (bottom squared) portion of the equation. I am unsure how to proceed.

Many thanks,

Nathaniel

2. First of all, if $\displaystyle u = 5(2x^2 + 3)^3$ then $\displaystyle u' = 60x(2x^2 + 3)^2$

and if $\displaystyle v = 3(4x^3 - 1)^2$ then $\displaystyle v' = 72x^2(4x^3 - 1)$.

3. Thanks Prove It. I'll make the modifications and perform the calculations again. But once I've done that, will I need to differentiate the (bottom squared) part before expanding?

4. UPDATE:

I think I've incorporated Prove It's corrections, and am again up to the last section of the equation where I need to take the squared bottom part of the equation into account.

Here are my current workings and where I am up to:

From there, how do I finish solving the equation? Do I need to differentiate the bottom, or expand it normally? Or is the answer correct as it is?

Many thanks,

Nathaniel