Question

знайдіть похідну пожалуйста
а) y=2x^3
б) y=1:x^6
в) y=x+1:x-1

Answer 1

$1.\ [u+v]'=u'+v'\\\\ 2.\ [x^a]'=a*x^{a-1}\\\\ 3.\ [C*f(x)]'=C*[f(x)]'=C*f'(x)\\\\4.\ [C]'=0$

$1.\ y'(x)=[2x^3]'=2*[x^3]'=2*3*x^{3-1}=6x^2\\\\ 2.\ y'(x)=[1:x^6]'=[\frac{1}{x^6}]'=[x^{-6}]'=-6*x^{-6-1}=\\\\=-6*x^{-7}=-6*\frac{1}{x^7}=-\frac{6}{x^7}\\\\ 3.\ y'(x)=[x+1:x+1]'=[x+\frac{1}{x}+1]'=[x^1+x^{-1}+1]=\\\\ =[x^1]'+[x^{-1}]'+[1]'=1*x^{1-1}+(-1)*x^{-1-1}+0=\\\\ =1*x^0-x^{-2}=1*1-\frac{1}{x^2}=1-\frac{1}{x^2}$