Question

The Product Rule
Which is more correct?
\[\left(fg\right)^\prime=f^\prime g+g^\prime f\]
or
\[\left(fg\right)^\prime=f^\prime g+f g^\prime \]

Answer 1

second one?

Answer 2

justify,

Answer 3

How can one be 'more' correct than the other one if they're the same?

Answer 4

they are not the same the ordering is different

Answer 5

The ordering is different but by commutative law of multiplication, they're the same.

Answer 6

can you assume commutativity
?

Answer 7

perhaps they are anti-commutative

Answer 8

Well, commutative law of multiplication states that
\[\forall a,b\in \mathbb{R}\ (ab=ba)\]
A function f from R to R (subsets of R will work too) goes like this:
\[\begin{align}
f:\mathbb{R}\ &\to \mathbb{R}\\
x\ &\mapsto f(x)
\end{align}\]
Thus, the assignment assigns a real number to another real number, which satisfies the commutative law of multiplication.

Answer 9

both will give same result!!

Answer 10

not all products are commutative

Answer 11

example??

Answer 12

Yeah, like the cross product.

Answer 13

matrices

Answer 14

this is not cross product and this is not matrix!!

Answer 15

\[f,g \cancel \in \mathbb R\]

Answer 16

Personally, I'd like to memorize the rule in the 2nd form, because it helps me facilitate the memorization of this rule:
\[\frac{d}{dt}[\mathbf{u}(t)\times\mathbf{v}(t)]=\mathbf{u}'(t)\times\mathbf{v}(t)+\mathbf{u}(t)\times\mathbf{v}'(t)\]

Answer 17

So how do the assignments f and g go, if they're not from R to R, such that the elements of the codomain won't satisfy commutativity?

Answer 18

doing it any vch way vl not distinct the answer!!

Answer 19

lol briefsssss

Answer 20

I think the most easy is this one:

\[\large (fg)' = fg' + gf'\]