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let-f-x-x-1-e-x-and-g-x-ln-2-x-2-1-calculate-fog-x-and-gof-x-2-calculate-fog-x-and-gof-x-

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Question Number 38722 by maxmathsup by imad last updated on 28/Jun/18
let f(x)= (x+1)e^(−x) and g(x)=ln(2+x^2 ) 1) calculate fog(x) and gof(x) 2) calculate (fog)^′ (x) and (gof)^′ (x).
letf(x)=(x+1)e−xandg(x)=ln(2+x2)1)calculatefog(x)andgof(x)2)calculate(fog)′(x)and(gof)′(x).
Commented by math khazana by abdo last updated on 30/Jun/18
1) fog(x)=f(g(x))=(g(x)+1)e^(−g(x)) ={ln(2+x^2 )+1} e^(−ln(2+x^2 )) = ((1+ln(2+x^2 ))/(2+x^2 )) gof(x)=g(f(x))= ln(2+f^2 (x)) =ln(2+(x+1)^2 e^(−2x) ) 2)(fog)^′ (x)=((((2x)/(2+x^2 ))(2+x^2 ) −2xln(2+x^2 ))/((2+x^2 )^2 )) = ((2x −2xln(2+x^2 ))/((2+x^2 )^2 )) . (gof)^′ (x) = ((2(x+1)e^(−2x) −2(x+1)^2 e^(−2x) )/(2+(x+1)^2 e^(−2x) )) =(({2x+2−2x^2 −4x−2}e^(−2x) )/(2+(x+1)^2 e^(−2x) )) =(((−2x^2 −2x)e^(−2x) )/(2+(x+1)^2 e^(−2x) )) = ((−2x^2 −2x)/(2 e^(2x) +(x+1)^2 )) .
1)fog(x)=f(g(x))=(g(x)+1)e−g(x)={ln(2+x2)+1}e−ln(2+x2)=1+ln(2+x2)2+x2gof(x)=g(f(x))=ln(2+f2(x))=ln(2+(x+1)2e−2x)2)(fog)′(x)=2x2+x2(2+x2)−2xln(2+x2)(2+x2)2=2x−2xln(2+x2)(2+x2)2.(gof)′(x)=2(x+1)e−2x−2(x+1)2e−2x2+(x+1)2e−2x={2x+2−2x2−4x−2}e−2x2+(x+1)2e−2x=(−2x2−2x)e−2x2+(x+1)2e−2x=−2x2−2x2e2x+(x+1)2.

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