My brain Gave up on this problem....
Find the derivative:
$\displaystyle D_x(3x^3 + x^{-3})(x+3)(x^2-5)$

My brain Gave up on this problem....
Find the derivative:
$\displaystyle D_x(3x^3 + x^{-3})(x+3)(x^2-5)$
Use the product rule:

$\displaystyle D_x(3x^3 + x^{-3})(x+3)(x^2-5)=$$\displaystyle D_x[3x^3+x^{-3}]\ (x+3) (x^2-5) + (3x^3+x^{-3}) D_x[(x+3)(x^2-5)]$

......$\displaystyle =(9x^2-3x^{-4})(x+3)(x^2-5) +(3x^3+x^{-3}) [(x^2-5)+(x+3)(2x)]$

I will leave simplifying this further to you.

RonL

$\displaystyle 18x^5+45x^4-60x^3-120x^2-3x^{-2}-5x^{-3}$

Find the derivative: .$\displaystyle y \:=\:(3x^3 + x^{-3})(x+3)(x^2-5)$

There is an extended product rule . . .

If $\displaystyle y \:=\:f\!gh$, then: .$\displaystyle y' \;=\;f'gh + f\!g'h + f\!gh'$

If $\displaystyle y \:=\:f\!ghk$, then: .$\displaystyle y' \;=\;f'ghk + f\!g'hk + f\!gh'k + f\!ghk'$

. . Get the pattern?

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

We have: .$\displaystyle y \;=\;3x^6 + 9x^5 - 15x^4 - 45x^3 + 1 - 3x^{-1} - 5x^{-2} - 15x^{-3}$
Then: .$\displaystyle y' \;=\;18x^5 + 45x^4 - 60x^3 - 135x^2 + 3x^{-2} + 10x^{-3} + 45x^{-4}$