Q.Find the $n^{th}$ order derivative of $\log x$.

For most functions there is no single formula that works for every function's $n$th derivative — each standard function has its own pattern that must be discovered by repeated differentiation. The method: differentiate the function a first, second, and third time, write out each result, and look for a pattern in the coefficients, the power of $x$, the angle, or the sign that changes from one derivative to the next; once the pattern is clear, express it directly as a formula in $n$. This produces standard results worth knowing: $\dfrac{d^n}{dx^n}(x^m)=\dfrac{m!}{(m-n)!}x^{m-n}$ when $m\ge n$; $\dfrac{d^n}{dx^n}\!\left[\dfrac{1}{ax+b}\right]=(-1)^n n!\,a^n(ax+b)^{-n-1}$; $\dfrac{d^n}{dx^n}(\log x)=(-1)^{n-1}(n-1)!\,x^{-n}$; $\dfrac{d^n}{dx^n}(\sin x)=\sin\!\left(x+\dfrac{n\pi}{2}\right)$ and $\dfrac{d^n}{dx^n}[\c …