# Thread: n'th order rules of derivation

1. ## n'th order rules of derivation

since one can take the derivative of a function multiple times it would be ideal to developa formula for the n'th derivative of a function to avoid many steps for large n (if there were any reason to find the 10th order derivative)

for example:
$\displaystyle \frac{d^n}{dx^n}x^p=\frac{p!}{(p-n)!}x^{p-n}$
$\displaystyle \frac{d^n}{dx^n}a^x=\ln(a)^na^x$
$\displaystyle \frac{d^n}{dx^n}\sin(x)=\sin(x+n\frac{\pi}{2})$

also product rule becomes

$\displaystyle \frac{d^n}{dx^n}f(x)g(x)=\sum_{k=0}^{n}(_{n}C_{k}) \frac{d^{n-k}}{dx^{n-k}}f\frac{d^k}{dx^k}g$

Can similar rules be established for the chain rule $\displaystyle \frac{d^n}{dx^n}f(g(x))$ and for the quotient (and/or reciprocal rule) $\displaystyle \frac{d^n}{dx^n}\frac{f(x)}{g(x)}$ (and/or $\displaystyle \frac{d^n}{dx^n}\frac{1}{f(x)}$)? What are they?

2. I think I have seen one for the chain rule before (I may be wrong), if I did, then it was extremely messy. As for the quotient rule, if you let $\displaystyle h(x) = \frac{1}{g(x)}$ then $\displaystyle \frac{d^n}{dx^n} \frac{f(x)}{g(x)} = \frac{d^n}{dx^n} f(x) h(x) = \sum_{k=0}^{n}(_{n}C_{k})\frac{d^{n-k}}{dx^{n-k}}f\frac{d^k}{dx^k}h$

The reciprocal rule is just a special case of the quotient rule, where the function in the numerator is identically equal to 1, so the above formula covers that as well.

3. For the chain rulie formula you want, check out this wiki page:

Faà di Bruno's formula - Wikipedia, the free encyclopedia

4. Originally Posted by JG89
I think I have seen one for the chain rule before (I may be wrong), if I did, then it was extremely messy. As for the quotient rule, if you let $\displaystyle h(x) = \frac{1}{g(x)}$ then $\displaystyle \frac{d^n}{dx^n} \frac{f(x)}{g(x)} = \frac{d^n}{dx^n} f(x) h(x) = \sum_{k=0}^{n}(_{n}C_{k})\frac{d^{n-k}}{dx^{n-k}}f\frac{d^k}{dx^k}h$

The reciprocal rule is just a special case of the quotient rule, where the function in the numerator is identically equal to 1, so the above formula covers that as well.
the proof of the reciprocal rule should be needed to prove the quotient rule for the reasons you stated. However a reciprocal rule is needed to evalute the nth order derivative of a function such as cosecant. the nth order dervatives of sine are known and I would like to express the nth order derivative of cosecant interms of a finite number of algebraic and trigonometric expressions involving the various derivatives of sine. (the whole idea ame up while trying to take the fractional derivative of 'tan(x)')

5. Originally Posted by JeffN12345
the proof of the reciprocal rule should be needed to prove the quotient rule for the reasons you stated. However a reciprocal rule is needed to evalute the nth order derivative of a function such as cosecant. the nth order dervatives of sine are known and I would like to express the nth order derivative of cosecant interms of a finite number of algebraic and trigonometric expressions involving the various derivatives of sine. (the whole idea ame up while trying to take the fractional derivative of 'tan(x)')
quotient rule for higher order derivatives

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### n th darivation

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