Question

f:(-2,+infinit)->R,f(x)=xe^x/x+2.
Demonstrati ca f'(x)=(x^2+2x+2)e^x/(x+2)^2,x apartine (-2,+infinit).

Answer 1

( x * e^x ) ' = e^x + x*e^x = e^x( 1 + x ) ;
( x + 2 )' = 1 ;
( u / v ) ' = (u' * v - u * v' ) / v^2;
Aplici formulele de mai sus =>[ e^x( 1 + x ) *  ( x + 2 ) - x * e^x * 1 ] / ( x + 2 )^ 2 =
e^x( x + 2 + 2x + x^2 - x ) / ( x + 2 )^ 2 = ( x^2 + 2x + 2 ) / ( x + 2 ) ^2 , oricare ar fi x ∈ ( - 2 , +oo ) ;

Bafta !