# Thread: "Let f and g be twice differentiable functions ..." - Derivitive problem

1. ## "Let f and g be twice differentiable functions ..." - Derivitive problem

There's a problem that our professor wanted us to look at and I'd really like to understand it. It says

"Let f and g be twice differentiable functions. Derive a formula for (fg'') = [(fg)']' using your knowledge of differentiation rules."

We know about the power rule and chain rule and all that fun stuff .. I guess I'm not sure what she's asking.

Am I trying to apply these rules to "[(fg)']'" to make it look like (fg'')? If I am, I don't know how I would even start that.

Any help would be appreciated.

Thanks

$\displaystyle (fg)''=[(fg)']'=(f'g+g'f)'$

by the Product Rule. By linearity,

$\displaystyle (f'g+g'f)'=(f'g)'+(g'f)'.$

3. Originally Posted by JTG2003
There's a problem that our professor wanted us to look at and I'd really like to understand it. It says

"Let f and g be twice differentiable functions. Derive a formula for (fg'') = [(fg)']' using your knowledge of differentiation rules."

We know about the power rule and chain rule and all that fun stuff .. I guess I'm not sure what she's asking.

Am I trying to apply these rules to "[(fg)']'" to make it look like (fg'')? If I am, I don't know how I would even start that.

Any help would be appreciated.

Thanks
your notation should be $\displaystyle (fg)'' = [(fg)']'$

$\displaystyle (fg)' = fg' + f'g$

$\displaystyle [(fg)']' = (fg' + f'g)' = fg'' + f'g' + f'g' + f''g = fg'' + 2(f'g') + f''g$

so ...

$\displaystyle (fg)'' = [(fg)']' = fg'' + 2(f'g') + f''g$

4. Hm... ok, I think I get it.

It will take a few times of going through it I guess.

Thank you.