Higher Order Derivatives

1.

Leibnitz's formula

$\left(uv\right)^{\prime\prime}=u^{\prime\prime} v + 2u^\prime v^\prime + u v^{\prime\prime}$

2.

Third derivative of a product

$\left(uv\right)^{\prime\prime\prime} = u^{\prime\prime\prime}v + 3u^{\prime\prime}v^\prime + 3u^\prime v^{\prime\prime} + 3uv^{\prime\prime\prime}$

3.

n-th derivativative of $x^m$

$\left(x^m\right)^{(n)}=\dfrac{m!}{(m-n)!}x^{m-n}$

4.

n-th derivative of $x^n$

$\left(x^n\right)^{(n)}=n!$

5.

n-th derivative of logarithm

$\left( \log_a x\right)^{(n)}=\dfrac{(-1)^{n-1} (n-1)! }{x^n \ln x}$

6.

n-th derivative of natural logarithm

$\left(\ln x \right)^{(n)}=\dfrac{(-1)^{n-1} (n-1)!}{x^n}$

7.

n th derivative od $a^x$

$\left(a^x\right)^{(n)}=a^x \ln^n a$

8.

n-th derivative of $e^x$

$\left(e^x\right)^{(n)}=e^x$

9.

$\left( a^{mx} \right)^{(n)} = m^n a^x \ln^na$

10.

n-th derivativ of sin x

$\left( \sin x \right)^{(n)} = \sin\left(x+\dfrac{n\pi}{2}\right)$

11.

n-th derivative od cos x

$\left( \cos x \right)^{(n)} = \cos\left(x+\dfrac{n\pi}{2}\right)$