Question

prove that given f, g differentiable blah blah that
\[(fg)'=f'g+g'f\] using only that
\[(f^2)'=2ff'\]

Answer 1

so why can't we use definition of a derivative lol

Answer 2

you can but then you have to do that stupid non intuitive add and subtract business, when in fact it is only algebra

Answer 3

because i'm only allowed to use (f^2)'=2ff'

Answer 4

yes only the rule for squares which easy to prove.

Answer 5

so we have (f^2)'=2ff'=ff'+ff'...
so im thinking

Answer 6

maybe look at ((f+g)^2)'

Answer 7

can we use (f+g)'=f'+g'?

Answer 8

yes

Answer 9

ok well then its done

Answer 10

nice way of doing it

Answer 11

i thought so because the usual method makes more out of it than there really is

Answer 12

[(f+g)^2]'
=2(f+g)(f+g)'
=2(f+g)(f'+g')
=2(ff'+fg'+gf'+gg')
=2ff'+2fg'+2gf'+2gg'
=(f^2)'+2fg'+2gf'+(g^2)'
[(f+g)^2]'-(f^2)'-(g^2)'=2(fg'+gf')
[(f+g)^2-f^2-g^2]'=2(fg'+gf')
[f^2+2fg+g^2-f^2-g^2]'=2(fg'+gf')
[2fg]'=2(fg'+gf')
(fg)'=fg'+gf'

Answer 13

i like this satellite

Answer 14

lol i like it too

Answer 15

usually start with
\[(f+g)^2=f^2+g^2+2fg\]
\[fg=\frac{1}{2}((f+g)^2-f^2-g^2)\] which also says that any product can be written as the sum and difference of squares

Answer 16

i never thought to prove the product rule this way

Answer 17

thanks for new knowledge

Answer 18

yep...good stuff!

Answer 19

i do love using the definition of derivative though

Answer 20

next semester...
if you start with
\[fg=\frac{1}{2}((f+g)^2-f^2-g^2)\] puts all the computation on one side of the equal sign, easier for students to follow usually

Answer 21

no it won't help lol
nothing will

Answer 22

ask them do it themselves. the algebra i mean.

Answer 23

lol

Answer 24

i already give up them and haven't even met them

Answer 25

give up on them*

Answer 26

i might ask my cal 2 students to do it on a test

Answer 27

they don't even know how to differentiate so it will be good for them

Answer 28

now you sound like a math teacher...

Answer 29

it is generally good to assume little or no knowledge and start from scratch. at least where i am

Answer 30

i thought i sounded like an english teacher

Answer 31

lol!

Answer 32

try 3 one hour twenty minute classes in a row! not enough americanos in the whole coffee shop to get me through