# Thread: n Order Derivative Quotient Problem

1. ## n Order Derivative Quotient Problem

$\displaystyle f^{(n)}(a)$, $\displaystyle g^{(n)}(a)$, their nth ord. derivatives exist and are not zero then $\displaystyle \Big(\frac{f}{g}\Big)^{(n)} = \frac{g(a)f^{(n)}(a) + (-1)^nf(a)g^{(n)}(a)}{g^{(n+1)}(a)}$. So, how should I start off? Should I let $\displaystyle h(x):= \frac{1}{g}$ then we have $\displaystyle \Big(\frac{f}{g}\Big)^{(n)} = (f(x)\cdot h(x))^{(n)}$ Or is there a better way to approach it? Note: this problem could be false.

2. Originally Posted by Endowed
$\displaystyle f^{(n)}(a)$, $\displaystyle g^{(n)}(a)$, their nth ord. derivatives exist and are not zero then $\displaystyle \Big(\frac{f}{g}\Big)^{(n)} = \frac{g(a)f^{(n)}(a) + (-1)^nf(a)g^{(n)}(a)}{g^{(n+1)}(a)}$. So, how should I start off? Should I let $\displaystyle h(x):= \frac{1}{g}$ then we have $\displaystyle \Big(\frac{f}{g}\Big)^{(n)} = (f(x)\cdot h(x))^{(n)}$ Or is there a better way to approach it? Note: this problem could be false.
Check to see if it's true, but try induction.