Given that $f(x) = \frac{1}{x}$, write an expression for $f^{(n)}(x)$ in terms of x and n.
The first part of the question is to find the first four derivatives of $f(x)$, which I got: $$-x^{-2}, 2x^{-3}, -6x^{-4} \text{ and }\space 24x^{-5}$$ The answer is $\dfrac{(-1)^n * n!}{x^{n-1}}$, but I have no idea how to get to that solution.
Please help with a thorough step-by-step if you can!
Given that they tell you to find the first four derivatives, they are hinting at you to find the pattern.
Getting the $(-1)^n$ out of the way first (I trust you see that part pretty quickly), we can tackle the coefficient next.
Taking 1, 2, 6, 24 and noticing right away it is not geometric, and there definitely isn't some constant being added, and maybe appealing to some experience with things like taylor series, we might guess that this is a factorial term. Checking, we see that $1=1!, 2=2!, 3=3!, 4=4!$ so our guess seems to work.
Finally, we figure out the powers of x: $n=1, x^{-2}$ $n=2, x^{-3}$ $n=3, x^{-4}$ $n=4, x^{-5}$
Looks like $x^{-(n+1)}=x^{-n-1}$ works,yielding our expression.
