Question Number 134884 by abdullahquwatan last updated on 08/Mar/21
$$\mathrm{if}\:\mathrm{2x}^{\mathrm{3}} −\mathrm{2}=\int_{{a}} ^{\mathrm{x}} \mathrm{f}\left(\mathrm{t}\right)\mathrm{dt},\:\mathrm{then}\:\mathrm{f}\:'\left({a}\right)=... \\ $$
Answered by bemath last updated on 08/Mar/21
![(d/dx) [ 2x^3 −2 ]= f(x) ⇒ f(x)= 6x^2 ; f ′(x)=12x ⇒f ′(a)=12a](Q134887.png)
$$\:\frac{\mathrm{d}}{\mathrm{dx}}\:\left[\:\mathrm{2x}^{\mathrm{3}} −\mathrm{2}\:\right]=\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\:\Rightarrow\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{6x}^{\mathrm{2}} \:;\:\mathrm{f}\:'\left(\mathrm{x}\right)=\mathrm{12x} \\ $$$$\Rightarrow\mathrm{f}\:'\left({a}\right)=\mathrm{12}{a} \\ $$
Commented by abdullahquwatan last updated on 08/Mar/21
$$\mathrm{thx}\:\mathrm{sir} \\ $$
Commented by bemath last updated on 08/Mar/21
$$\mathrm{you}\:\mathrm{put}\:{a}\:=\:\mathrm{1} \\ $$
Answered by Ñï= last updated on 08/Mar/21
$${f}\left({x}\right)=\mathrm{6}{x}^{\mathrm{2}} \\ $$$${f}\left({x}\right)'=\mathrm{12}{x} \\ $$$$\int_{{a}} ^{{x}} {f}\left({t}\right){dt}=\int_{{a}} ^{{x}} \mathrm{6}{x}^{\mathrm{2}} {dx}=\mathrm{2}{x}^{\mathrm{3}} −\mathrm{2}{a}^{\mathrm{3}} =\mathrm{2}{x}^{\mathrm{3}} −\mathrm{2} \\ $$$$\Rightarrow{a}^{\mathrm{3}} =\mathrm{1} \\ $$$$\Rightarrow{f}\left({a}\right)'={e}^{\mathrm{2}{ik}\pi/\mathrm{3}} \:\:\:\:\:\:\left({k}=\mathrm{0},\mathrm{1},\mathrm{2}\right) \\ $$