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Math Help - "Let f and g be twice differentiable functions ..." - Derivitive problem

  1. #1
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    "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
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  2. #2
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    To start with, you may note that

    (fg)''=[(fg)']'=(f'g+g'f)'

    by the Product Rule. By linearity,

    (f'g+g'f)'=(f'g)'+(g'f)'.
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  3. #3
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    Quote Originally Posted by JTG2003 View Post
    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 (fg)'' = [(fg)']'


    (fg)' = fg' + f'g

    [(fg)']' = (fg' + f'g)' = fg'' + f'g' + f'g' + f''g = fg'' + 2(f'g') + f''g

    so ...

    (fg)'' = [(fg)']' = fg'' + 2(f'g') + f''g
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  4. #4
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    Hm... ok, I think I get it.

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

    Thank you.
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