Table of Higher Order Derivatives

\((x^p)^{(n)}= p(p-1)(p-2)…(p-n+1)x^{p-n}.\)

\( (a^x)^{(n)}=a^x\ln^na \qquad\qquad(e^x)^{(n)}=e^x\)

\((\sin \alpha x)^{(n)}=\alpha^n\sin\left(\alpha x+\frac{\pi n}{2}\right)\)

\((\cos \alpha x)^{(n)}=\alpha^n\cos\left(\alpha x+\frac{\pi n}{2}\right)\)

\(\left((ax+b)^p\right)^{(n)}=a^np(p-1)(p-2)…(p-n+1)(ax+b)^{p-n}\)

\((\log_a |x|)^{(n)}=\frac{(-1)^{n-1}(n-1)!}{x^n\ln a}\)

\((\ln |x|)^{(n)}=\frac{(-1)^{n-1}(n-1)!}{x^n}\)

\((\alpha u(x)+\beta v(x))^{(n)}=\alpha u^{(n)}(x)+\beta v^{(n)}(x)\)

\((u(x)v(x))^{(n)}=\sum\limits_{k=0}^n C_n^k u^{(k)}(x)v^{(n-k)}(x),\,\, \mbox {where}\quad C_n^k=\frac{n!}{k!(n-k)!}\)

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