Question

Please help, don't understand !
Show that
[D(fg)(a)]x = (Df(a)x)(g(a)) + (Dg(a)x)(f(a))

The hint is to try x = e1 and x = e2, but idk how to go about showing that .. matrix way, or .. ?

Answer 1

what does [D(fg)(a)]x represent?

Answer 2

the derivative of the product of fg multiplied by a vector x

Answer 3

at some point "a"? or is "a" variable?

Answer 4

it is differentiable at some vector a belonging to Rn and x belongs to Rn

Answer 5

On my sheet theres an arrow over the a, so I'm assuming its a vector as well

Answer 6

\[[D(fg)(\vec a)]\vec x=D(f(\vec a)\vec x)g(\vec a)+D(g(\vec a)\vec x)f(\vec a)~~~~~~~~~\vec a,\vec x\in\mathbb R^n\]like so?

Answer 7

yes, except on my sheet, for the last part the f(a) is before the D(g(a))x

Answer 8

I'm not sure if that matters .. but for the second part of the question matrices are involved, so I'm assuming it does ..

Answer 9

\[[D(fg)(\vec a)]\vec x=D(f(\vec a)\vec x)g(\vec a)+f(\vec a)D(g(\vec a)\vec x)~~~~~~~~~\vec a,\vec x\in\mathbb R^n\]

Answer 10

yes, and f,g : Rn -> R3

Answer 11

seems like a toughie...

Answer 12

I'm not really sure how to write what happens if you plug in \(x=\vec e_1\)

Answer 13

i know .. it just says consider the cases when x = e1, x = e2 ..

Answer 14

I may or may not come up with something, but I'll be thinking about it...

Answer 15

thank you, appreciate it.