Nth derivative of a function: I don't know where to start

I'm practicing some problems from past exams and found this one: Find nth derivative of this function:
$f(x)=\frac {x} {x^2-1}$.
I have no idea how to start solving this problems. Is there any theorem for finding nth derivative?

Solution 1:

Maybe we can add some more help -just in case you didn't succeed to find the answer yourself yet. Let

$$ \frac{x}{x^2 - 1} = \frac{A}{x-1} + \frac{B}{x+1} $$

be the splitting into partial fractions. (I'm too lazy to compute the coeffitients $A$ and $B$.) Then

$$ \frac{d}{dx} \frac{x}{x^2 - 1} = -\frac{A}{(x-1)^2} - \frac{B}{(x+1)^2} \ . $$

Differentiating again,

$$ \frac{d^2}{dx^2}\frac{x}{x^2 - 1} = \frac{2A}{(x-1)^3} + \frac{2B}{(x+1)^3} \ . $$

One more time:

$$ \frac{d^3}{dx^3} \frac{x}{x^2 - 1} = - \frac{3\cdot 2 A}{(x-1)^4} - \frac{3\cdot 2 B}{(x+1)^4} \ . $$

And sure enough you can find the general pattern now, can't you? Then, use induction to prove your guess.

Solution 2:

HINT $\;\;\;$ Upon employing partial fractions it reduces to $\;\;\;\rm D^n \:\frac{1}{x+1}. \;\;$ Now employ the Taylor series

$$\rm\; f(t+x) = \sum_{k=0}^\infty \;\; D^n \: f(x) \; \frac{t^n}{n!}$$

and note that for this problem we've $\rm\; f(t+x) = \frac{1}{t+x+1}$ is a geometric series with known coefficients.

Such "generating function" approaches often work smoothly even in much more complicated problems. Indeed, there is a very powerful Umbral calculus that frequently succeeds in computing such closed form expressions, e.g. see Steven Roman: The Umbral Calculus. 1984. For example, below is a small sample of derivative formulas for the countless number of polynomial sequences amenable to umbral calculus analysis

$$\begin{array}{|r|l|} \hline \rm Name & \rm Derivative \; formula \\ \hline\hline \rm Laguerre & \rm L_n^k(x) = (D+1)^{n+k}(-x)^n \\ \rm Exponential & \rm\;\; e_n(x) = e^{-x}(xD)^n e^x \\ \rm Abel & \rm A_n^k(x) = x \; e^{-knD} x^{n-1} \\ \rm Hermite & \rm H_n^k(x) = (-1)^n e^{x^2/(2n)} (kD)^n e^{-x^2/(2n)} \\ \rm Bernoulli & \rm B_n^k(x) = \left(\frac{D}{e^D-1}\right)^k x^n \\ \rm Euler & \rm E_n^k(x) = \left(\frac{2}{e^D+1}\right)^k x^n \\ \end{array}$$

Solution 3:

Split it into partial fractions then differentiate.

Solution 4:

To add to Derek's hint: you will have to show the validity of the formula

$\frac{\mathrm{d}^k}{\mathrm{d}x^k}\frac1{x}=\frac{(-1)^k k!}{x^{k+1}}$