# Thread: Finding derivatives with ln

1. ## Finding derivatives with ln

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

How do I find that?

2. just differentiate it 9 times using the chain rule

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

there is a pattern for this derivatives

3. Originally Posted by arkhampatient
$\displaystyle \frac{d^9}{dx^9} (x^8 ln x)$

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

4. Originally Posted by arkhampatient
$\displaystyle \frac{d^9}{dx^9} (x^8 ln x)$

How do I find that?
Just apply the product rule

5. So with each successive derivation I keep adding another term

My answer is a value is that right?

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

I keep expanding it then leads me to this:

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

Am I doing something wrong?