Solve using quotient rule y=5x^3/(4x^2-3)^3

I have manage to differentiate the question but I’ve got completely confused when trying to simplify the answer.
I have factored out the common factor but I’ve got a little confused on the way as to what to do next

Can you post a pic of what youve done? It shouldnt be too bad if you factor out the common parts correctly.

Sorry, i had started the message but got distracted and forgot click update

You can cancel the (4x^2-3)^2 on the numerator with (part of) the denominator. But your last term (4x^2-3)-8x^2 on the penultimate line going to the last line isnt correct. Its subtraction, not multiplication.

You mean like this? I did that previously but leaving the other 4x²-3 didn’t seem right to me.

Yes, now simplify the [...] on the numerator.

Sorry, I took a break and started reading last night.

So, expand like so, and simply fly by extracting the common factor -8x² and multiplying by 15x²?
So, that would be -120x⁴(4x²-3)/(4x²-3)⁴ ?

As a side note, obviously if the question says "use the quotient rule", you have no choice, but if you have something of the form $\dfrac{d}{dx}\dfrac{f(x)}{g(x)^\alpha}$ (with $\alpha \neq 1$), it's generally easier to rewrite this as $\dfrac{d}{dx}f(x)g(x)^{-\alpha}$ and use the product rule.
[they're not really *that* different, but you don't end up with as much to cancel at the simplification stage].

Youre subtracting 8x^2, not multiplying by -8x^2.

As a side note, obviously if the question says "use the quotient rule", you have no choice, but if you have something of the form $\dfrac{d}{dx}\dfrac{f(x)}{g(x)^\alpha}$ (with $\alpha \neq 1$), it's generally easier to rewrite this as $\dfrac{d}{dx}f(x)g(x)^{-\alpha}$ and use the product rule.
[they're not really *that* different, but you don't end up with as much to cancel at the simplification stage].

Oh, poop!! that must have been where I’ve been going wrong!!
Okay, so: Late reply: partner wanted to go shopping for her birthday

Id have taken just the - out and left it factorised so
-15x^2(...)/(...)
but otherwise fine.

So, leave as it is but just take out the negative. Thank you. It’s annoying sometimes when I advance a lot and then recall so many formulas but forget how to do the simple maths that I learned way back.

note the edit about the denom typo.

Edit: I just realised what is happening here, you’re just multiplying the denominator with the numerator which explains the negative power and why chain rule would work here. Which I now see why the chain rule is the go to, over using quotient rule.

As you’re both here, could you help me understand what is the fundamental of calculus is? I’ve briefly learned that it’s what connects differentiation with integration but I don’t know the full concept.