Suppose-function-f-R-R-with-f-2x-3-4x-2-2x-5-and-f-denote-is-derivatif-of-f-function-What-the-value-of-f-2x-3-

Question Number 133616 by benjo_mathlover last updated on 23/Feb/21

$$\mathrm{Suppose}\:\mathrm{function}\:\mathrm{f}:\mathrm{R}\rightarrow\mathrm{R}\:\mathrm{with} \\ $$$$\mathrm{f}\left(\mathrm{2x}−\mathrm{3}\right)\:=\:\mathrm{4x}^{\mathrm{2}} +\mathrm{2x}−\mathrm{5}\:\mathrm{and}\:\mathrm{f}\:'\:\mathrm{denote} \\ $$$$\mathrm{is}\:\mathrm{derivatif}\:\mathrm{of}\:\mathrm{f}\:\mathrm{function}. \\ $$$$\mathrm{What}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\:'\left(\mathrm{2x}−\mathrm{3}\right) \\ $$

Commented by mr W last updated on 23/Feb/21

$$\mathrm{4}{x}+\mathrm{1} \\ $$

Answered by TheSupreme last updated on 23/Feb/21

D(f(g(x)))=g′(x)f′(g(x))=8x+2 g(x)=2x−3 → g′(x)=2 f′(g(x)))=4x+1

$${D}\left({f}\left({g}\left({x}\right)\right)\right)={g}'\left({x}\right){f}'\left({g}\left({x}\right)\right)=\mathrm{8}{x}+\mathrm{2} \\ $$$${g}\left({x}\right)=\mathrm{2}{x}−\mathrm{3}\:\rightarrow\:{g}'\left({x}\right)=\mathrm{2} \\ $$$$\left.{f}'\left({g}\left({x}\right)\right)\right)=\mathrm{4}{x}+\mathrm{1} \\ $$

Answered by liberty last updated on 23/Feb/21

⇒f(x)=4(((x+3)/2))^2 +2(((x+3)/2))−5 f(x)=x^2 +6x+9+x+3−5 f(x)=x^2 +7x+7 ⇒f ′(x)= 2x+7 f ′(2x−3)=2(2x−3)+7 = 4x+1

$$\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=\mathrm{4}\left(\frac{\mathrm{x}+\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{2}\left(\frac{\mathrm{x}+\mathrm{3}}{\mathrm{2}}\right)−\mathrm{5} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} +\mathrm{6x}+\mathrm{9}+\mathrm{x}+\mathrm{3}−\mathrm{5} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} +\mathrm{7x}+\mathrm{7}\: \\ $$$$\Rightarrow\mathrm{f}\:'\left(\mathrm{x}\right)=\:\mathrm{2x}+\mathrm{7} \\ $$$$\mathrm{f}\:'\left(\mathrm{2x}−\mathrm{3}\right)=\mathrm{2}\left(\mathrm{2x}−\mathrm{3}\right)+\mathrm{7}\:=\:\mathrm{4x}+\mathrm{1}\: \\ $$

Answered by mr W last updated on 23/Feb/21

let u=2x−3 ⇒(dx/du)=(1/2) f(u) = 4x^2 +2x−5 f′(u)=(df/du)=(df/dx)×(dx/du)=(8x+2)(1/2)=4x+1 i.e. f′(2x−3)=4x+1

$${let}\:{u}=\mathrm{2}{x}−\mathrm{3} \\ $$$$\Rightarrow\frac{{dx}}{{du}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{f}\left({u}\right)\:=\:\mathrm{4x}^{\mathrm{2}} +\mathrm{2x}−\mathrm{5} \\ $$$${f}'\left({u}\right)=\frac{{df}}{{du}}=\frac{{df}}{{dx}}×\frac{{dx}}{{du}}=\left(\mathrm{8}{x}+\mathrm{2}\right)\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{4}{x}+\mathrm{1} \\ $$$${i}.{e}.\:{f}'\left(\mathrm{2}{x}−\mathrm{3}\right)=\mathrm{4}{x}+\mathrm{1} \\ $$

Facebook Tweet Pin

Suppose-function-f-R-R-with-f-2x-3-4x-2-2x-5-and-f-denote-is-derivatif-of-f-function-What-the-value-of-f-2x-3-

Leave a Reply Cancel reply