Variables

f, u, v
→
Functions
n
→
Integer
1.
Leibnitz's formula
(uv)′′=u′′v+2u′v′+uv′′\left(uv\right)^{\prime\prime}=u^{\prime\prime} v + 2u^\prime v^\prime + u v^{\prime\prime}
Differentiation
2.
Third derivative of a product
(uv)′′′=u′′′v+3u′′v′+3u′v′′+3uv′′′\left(uv\right)^{\prime\prime\prime} = u^{\prime\prime\prime}v + 3u^{\prime\prime}v^\prime + 3u^\prime v^{\prime\prime} + 3uv^{\prime\prime\prime}
Differentiation
3.
n-th derivativative of xmx^m
(xm)(n)=m!(m−n)!xm−n\left(x^m\right)^{(n)}=\dfrac{m!}{(m-n)!}x^{m-n}
Differentiation
4.
n-th derivative of xnx^n
(xn)(n)=n!\left(x^n\right)^{(n)}=n!
Differentiation
5.
n-th derivative of logarithm
(log⁡ax)(n)=(−1)n−1(n−1)!xnln⁡x\left( \log_a x\right)^{(n)}=\dfrac{(-1)^{n-1} (n-1)! }{x^n \ln x}
Differentiation
6.
n-th derivative of natural logarithm
(ln⁡x)(n)=(−1)n−1(n−1)!xn\left(\ln x \right)^{(n)}=\dfrac{(-1)^{n-1} (n-1)!}{x^n}
Differentiation
7.
n th derivative od axa^x
(ax)(n)=axln⁡na\left(a^x\right)^{(n)}=a^x \ln^n a
Differentiation
8.
n-th derivative of exe^x
(ex)(n)=ex\left(e^x\right)^{(n)}=e^x
Differentiation
9.
(amx)(n)=mnaxln⁡na\left( a^{mx} \right)^{(n)} = m^n a^x \ln^na
Differentiation
10.
n-th derivativ of sin x
(sin⁡x)(n)=sin⁡(x+nπ2)\left( \sin x \right)^{(n)} = \sin\left(x+\dfrac{n\pi}{2}\right)
Differentiation
11.
n-th derivative od cos x
(cos⁡x)(n)=cos⁡(x+nπ2)\left( \cos x \right)^{(n)} = \cos\left(x+\dfrac{n\pi}{2}\right)
Differentiation

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