Question

qual seria a derivada sucessiva n=3 de f(x)=Ln (4x)

Answer 1

$Fun\c{c}\~ao:\\\\ f_{(x)}=\ln(4x)= \ln(4) + \ln(x)\\\\\\ Para\ n=1:\\\\ f'_{(x)}=(\ln(4))' + (\ln(x))'\\\\ f'_{(x)}=0+\dfrac{1}{x}\\\\ \boxed{f'_{(x)}=\dfrac{1}{x}=x^{-1}}\\\\\\\\ Para\ n=2\ (aplicando\ regra\ do\ tombo):\\\\ f''_{(x)}=(-1)x^{-1-1}\\\\ \boxed{f''_{(x)}=-x^{-2}=-\dfrac{1}{x^2}}\\\\\\\\ Para\ n=3\ (aplicando\ regra\ do\ tombo):\\\\ f'''_{(x)}=-(-2)x^{-2-1}\\\\ \boxed{f'''_{(x)}=2x^{-3}=\dfrac{2}{x^3}}$


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