Question

Sa se calculeze derivatele funcțiilor : f(x)=xe la x , f(x)=e la x sin x , f(x)=x la a 2 ctgx , f(x)= e la x (x-lnx)

Answer 1

$\\ \boxed{\left(f\cdot g\right)'=f'\cdot g+f\cdot g'} \\ \\ 1.) \left(xe^x\right) = 1\cdot \:e^x+e^xx = e^x+e^xx \\ \\ 2.) (e^x sin x)' = e^x\sin \left(x\right)+\cos \left(x\right)e^x \\ \\ 3.) (x^2 ctg x)' = 2x\cot \left(x\right)+\left(-\csc ^2\left(x\right)\right)x^2 = -x^2\csc ^2\left(x\right)+2x\cot \left(x\right) \\ \\ 4.) [e^x (xln x)]' =e^xx\ln \left(x\right)+\left(\ln \left(x\right)+1\right)e^x$