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Math Help - Finding derivatives with ln

  1. #1
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    Finding derivatives with ln

    \frac{d^9}{dx^9} (x^8 ln x)

    How do I find that?
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  2. #2
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    just differentiate it 9 times using the chain rule


    \frac{d}{dx} (x^8 ln x)=8x^7 ln x + x^7<br />

    there is a pattern for this derivatives
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  3. #3
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    Quote Originally Posted by arkhampatient View Post
    \frac{d^9}{dx^9} (x^8 ln x)

    How do I find that?
    this question is unlikely but i assume it is to test your skills in predicting patterns of differentiation. notice when you find y' you get 2 terms, y' you get 3 terrms and so on. also notice that the x^7 in the y' term will decrease by a degree of 1 every time you differentiate, but you are differentiating 9 times implying all the terms you generated through deriving that are a degree lower than 8 will become 0. you should see a pattern arise and realise that x^8 part will end up multiplying itself by 8 then 7 .... until it becomes a constant i.e 8! and the ln part will become 1/x.

    go figure the pattern out and see if you can understand how it becomes 8!/x
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  4. #4
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    Quote Originally Posted by arkhampatient View Post
    \frac{d^9}{dx^9} (x^8 ln x)

    How do I find that?
    Just apply the product rule
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  5. #5
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    So with each successive derivation I keep adding another term

    My answer is a value is that right?

    \frac {d^8}{dx^8} (8x^7 ln x + x^7)

    I keep expanding it then leads me to this:

    \frac {d}{dx} (40320 + 5040 + 720 + 120 + 24 + 12 + 3)

    Am I doing something wrong?
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