Question

f(x) = $\frac{1}{x}$
f^2019 ( x ) = ?
f^2019 = f derivat de 2019 ori

Answer 1

$f(x) = \dfrac{1}{x}\\ \\f(x) = x^{-1} \\ \\f^{(1)}(x) = -x^{-2} \\ \\ f^{(2)}(x) = 2x^{-3} \\ \\ f^{(3)} (x) = -2\cdot 3\cdot x^{-4} \\ \\f^{(4)}(x) = 2\cdot 3\cdot 4\cdot x^{-5}\\ --------\\ f^{(n)}(x) = (-1)^{n}\cdot n!\cdot x^{-n-1} \\ \\ \Rightarrow f^{(2019)}(x)=(-1)^{2019}\cdot 2019!\cdot x^{-2020} \\ \\ \Rightarrow \boxed{f^{(2019)}(x) = \dfrac{-2019!}{x^{2020}}}$