Successive Differentiation (nth Order Derivative) of Some Standard Functions

There is no single formula that gives the $n$-th derivative of every function — each standard function's own pattern has to be discovered individually. The method (usable with mathematical induction to make the final formula rigorous): Step 1, differentiate the function to get the 1st, 2nd and 3rd derivatives by ordinary rules. Step 2, line the three results up and observe how the coefficient, the sign, the power of $x$ (or, for a trig/exponential form, the phase angle added each time) changes from each derivative to the next. Step 3, express the $n$-th derivative directly from that observed pattern.

Worked Example 1 — find the nth derivative:

1. $y=x^m$: $\dfrac{dy}{dx}=mx^{m-1}$, $\dfrac{d^2y}{dx^2}=m(m-1)x^{m-2}$, $\dfrac{d^3y}{dx^3}=m(m-1)(m-2)x^{m-3}$ — each derivative multiplies by one less integer and drops the power by one, so in general $\dfrac{d^ny}{dx^n}=m(m-1)(m-2)\cdots[m-(n-1)]\,x^{m-n}.$ If $m$ is a positive integer with $m>n$, this can be written $\dfrac{d^ny}{dx^n}=\dfrac{m!}{(m-n)!}x^{m-n}$; if $m=n$, it equals $n!$ (a constant); if $m<n$, it is $0$ (every derivative beyond the $m$-th of a degree-$m$ polynomial vanishes).
2. $y=\dfrac1{ax+b}$: $\dfrac{dy}{dx}=\dfrac{-a}{(ax+b)^2}$, $\dfrac{d^2y}{dx^2}=\dfrac{2a^2}{(ax+b)^3}$, $\dfrac{d^3y}{dx^3}=\dfrac{-6a^3}{(ax+b)^4}$ — the sign alternates, the numerator coefficient is $n!$, and the denominator power is $n+1$: $\dfrac{d^ny}{dx^n}=\dfrac{(-1)^n\,n!\,a^n}{(ax+b)^{n+1}}.$
3. $y=\log x$: $\dfrac{dy}{dx}=\dfrac1x$, $\dfrac{d^2y}{dx^2}=-\dfrac1{x^2}$, $\dfrac{d^3y}{dx^3}=\dfrac2{x^3}$ — so $\dfrac{d^ny}{dx^n}=\dfrac{(-1)^{n-1}(n-1)!}{x^n}.$
4. $y=\sin x$: $\dfrac{dy}{dx}=\cos x=\sin\!\left(x+\dfrac\pi2\right)$; differentiating again shifts the angle by another $\dfrac\pi2$: $\dfrac{d^2y}{dx^2}=\sin\!\left(x+\dfrac{2\pi}2\right)$, $\dfrac{d^3y}{dx^3}=\sin\!\left(x+\dfrac{3\pi}2\right)$ — so $\dfrac{d^ny}{dx^n}=\sin\!\left(x+\dfrac{n\pi}2\right).$
5. $y=\cos(ax+b)$: each differentiation multiplies by $a$ and shifts the angle by $\dfrac\pi2$, exactly as in (iv), so $\dfrac{d^ny}{dx^n}=a^n\cos\!\left(ax+b+\dfrac{n\pi}2\right).$ …