Can you please tell me how to differentiate this? f(x) = x(x² - 3)(x³ + 1)?

Question

I'm just beginning to work with derivatives, and I've only done problems that are "1 step," and by that I mean, it's only multiplication, only division, only addition, etc. I'm not sure how to do this 2-step one using the product rule. So can you please explain as thoroughly as possible with WHAT you're doing and WHY you're doing it? THANK YOU SO MUCH!!!

Differentiate f(x) = x(x² - 3)(x³ + 1)

Answer 1

This can be done by simply multiplying everything out:

f(x) = x(x^5 - 3x^3 + x^2 - 3) = (x^6 - 3x^4 + x^3 - 3x)
d/dx(f(x)) = 6x^5 - 12x^3 + 3x^2 - 3

Or the product rule using u = x, v = (x² - 3) and w = (x³ + 1)

d/dx (uvw) = vw du/dx + uw dv/dx + uv dw/dx
du/dx = 1 , dv/dx = 2x , dw/dx = 3x^2

d/dx(f(x)) = (1)(x² - 3) (x³ + 1) +2x(x)(x³ + 1) + 3x^2(x)(x² - 3)

d/dx(f(x)) = x^5 - 3x^3 + x^2 - 3 + 2x^5 + 2x^2 + 3x^5 - 9x^3
d/dx(f(x)) = 6x^5 - 12x^3 + 3x^2 - 3

The product rule is that you simply go through each term and take the derivative, multiply it by the remaining terms and then add them all together.
d/dx (uv) = v du/dx + u dv/dx
d/dx (uvw) = vw du/dx + uw dv/dx + uv dw/dx
and so on

If you want to think of this in two steps, then define u, v and w as I did above and then use m=vw
So
d/dx(uvw) = d/dx(um) = m du/dx + u dm/dx
You then do dm/dx = d(vw)/dx = w dv/dx + v dw/dx
Put it all together and you get:
d/dx (uvw) = vw du/dx + uw dv/dx + uv dw/dx

Answer 2

f(x)=x(x^2-3)(x^3+1)
=(x^3-3x)(x^3+1) (multiply the first and second term)
use the product rule (you have two expressions)
let u=(x^3-3x)
v=(x^3+1)
then the derivative is udv+vdu (this is a formula to differenciate 2 expressions)
udv+vdu
=(x^3-3x)(3x^2) + (x^3+1)(3x^2-3) ---- (1)
=3x^2(x^3-3x) + (x^3+1)3(x^2-1)
=3x^2(x^3-3x) + 3(x^3+1)(x^2-1)
=3x^2(x^3-3x)+3(x^3+1)(x^2-1)
=3x^2(x^3-3x)+3(x^3+1)(x+1)(x-1)
The differenciation part is over at (1)
The rest are just simplifying the expression

Answer 3

x(x^2 - 3)(x^3 - 1)

Work out brackets....

x(x^5 - 3x^3 - x^2 +3)
= x^6 - 3x^4 - x^3 + 3x

Now differentiate.....

6x^5 - 12x^3 - 3x^2 + 3

Answer 4

use product rule, taking 2 terms together as one and the other one as one term, differentiate them again.