Question

Considere as funções f(x) = ln x/e^x e g(x) = ( ln x )^3 Calcule a derivada da soma f(x) + g(x) no ponto x = 1.

Answer 1

$h(x)=f(x)+g(x)\\ \\ f(x)=\frac{\ln x}{e^x}\\ \\ g(x)=(\ln^3 x)\\ \\ h'(x)=f'(x)+g'(x)\\ \\ f'(x)=\frac{(\frac1x)(e^x)-(e^x)(\ln x)}{e^{2x}}=\frac{1-x\ln x}{xe^x}\\\\g'(x)=3(\ln^2 x)(\frac1x)=\frac{3\ln^2x}{x}\\\\h'(x)=\frac{1-x\ln x}{xe^x}+\frac{3\ln^2x}{x}\\\\h'(1)=\frac{1-(1)\ln (1)}{(1)e^{(1)}}+\frac{3\ln^2(1)}{(1)}=\frac{1}{e}$