Question

найти производную функции при заданном значении аргумента

Answer 1

$f(x)=3sqrt{e^{4x+3}} \f'(x)=3cdot dfrac{1}{2sqrt{e^{4x+3}}} cdot (e^{4x+3})'=dfrac{3}{2sqrt{e^{4x+3}}} cdot e^{4x+3}cdot(4x+3)'=\\=dfrac{3e^{4x+3}}{2sqrt{e^{4x+3}}}cdot4=6sqrt{e^{4x+3}}\\f'(0)=6sqrt{e^{4cdot0+3}}=6sqrt{e^3}=6esqrt{e}$

$f(x)=lndfrac{x-1}{x^2+1} =ln(x-1)-ln(x^2+1)\\f'(x)=dfrac{1}{x-1} cdot(x-1)'-dfrac{1}{x^2+1}cdot(x^2+1)'=\\=dfrac{1}{x-1} cdot1-dfrac{1}{x^2+1}cdot2x=dfrac{1}{x-1}-dfrac{2x}{x^2+1}\\f'(2)=dfrac{1}{2-1}-dfrac{2cdot2}{2^2+1}=dfrac{1}{1}-dfrac{4}{4+1}=1-dfrac{4}{5}=dfrac{1}{5}$