Chain rule or product rule?

How would you differentiate y = 6x(1 + x^2 )^2

I'm not sure in what situation you use chain rule and in what situation you would use product rule?

If it was y = 6(1 + x^2 )^2
..could I use chain rule?

You would use both

Though, tbh, I would just multiply that out

What do you mean you would use both?

How would you differentiate y = 6x(1 + x^2 )^2

I'm not sure in what situation you use chain rule and in what situation you would use product rule?

If it was y = 6(1 + x^2 )^2
..could I use chain rule?

You can use whatever rule you like. $y(x) = 6(1+x^2)^2$ requires first the use of linearity of the derivative, to obtain $y'(x) = 6 \dfrac{d}{dx} (1+x^2)^2$, then you could use chain to get $y'(x) = 6 \times 2 (1+x^2) \dfrac{d}{dx} (1+x^2)$.

Alternatively, you could use product: $y(x) = 6 (1+x^2) (1+x^2)$ means $y'(x) = 6 (1+x^2) \dfrac{d}{dx}(1+x^2) + 6 (1+x^2) \dfrac{d}{dx} (1+x^2)$.

(By the way, $y(x) = 6x (1+x^2)^2 = \dfrac{d}{dx} (1+x^2)^3$ by chain rule. It doesn't help here, but some problems are made easier if you know that the function in question is the derivative of something.)

How would you differentiate y = 6x(1 + x^2 )^2

I'm not sure in what situation you use chain rule and in what situation you would use product rule?

If it was y = 6(1 + x^2 )^2
..could I use chain rule?

You seem to be differentiating a different question

Then you are referencing integration rules ... Why?

You can use whatever rule you like. $y(x) = 6(1+x^2)^2$ requires first the use of linearity of the derivative, to obtain $y'(x) = 6 \dfrac{d}{dx} (1+x^2)^2$, then you could use chain to get $y'(x) = 6 \times 2 (1+x^2) \dfrac{d}{dx} (1+x^2)$.

Alternatively, you could use product: $y(x) = 6 (1+x^2) (1+x^2)$ means $y'(x) = 6 (1+x^2) \dfrac{d}{dx}(1+x^2) + 6 (1+x^2) \dfrac{d}{dx} (1+x^2)$.

(By the way, $y(x) = 6x (1+x^2)^2 = \dfrac{d}{dx} (1+x^2)^3$ by chain rule. It doesn't help here, but some problems are made easier if you know that the function in question is the derivative of something.)

No he has not. Please read OP fully.

Also, he merely made a comment on the integration bit...noting that in this particular case it didnt help but can sometimes be useful in such a problem.

But when I differentiate y= 6x(1+ x^2)^2 using chain rule and the product rule I get different answers..

If you say so

There is no need for attitude. Please try to be more helpful on the maths forum in future

Once again ... You need to use both but it would be better to multiply it out


Can I check .... Since I have been told that I have misread

You are differentiating 6x(1+x^2)^2 and none of those x are a multiplication sign ... Or ?

Yes none of them are multiplication signs sorry.
So do you mean you would use chain regarding (1+x^2)^2 within the product rule?

No attitude here ... We clearly have a different understanding of this thread ... As for your second point ... Are you serious

Please lose the attitude. If you have nothing useful to contribute to help OP, then please dont at all

Exactly

Ah so you can't just use chain rule by itself to differentiate it... Thank you

There is no need for attitude. Please try to be more helpful on the maths forum in future