Question

Find f'(x). anything helps! also, bonus! f(x)= e^(2x-1) + e^2

Answer 1

$\bf f(x)=e^{-x}(e^{2x}+x^2)\iff f(x)=e^{-x}e^{2x}+e^{-x}x^2 \\\\\\ f(x)=e^{2x-x}+\cfrac{x^2}{e^x}\implies f(x)=e^x+\cfrac{x^2}{e^x}\\\\ -----------------------------\\\\ \cfrac{dy}{dx}\left[ \cfrac{x^2}{e^x} \right]\implies \cfrac{2xe^x-x^2e^x}{(e^x)^2}\implies \cfrac{xe^x(2-x)}{(e^x)(e^x)}\implies \cfrac{x(2-x)}{e^x}\\\\ -----------------------------\\\\ \cfrac{dy}{dx}=e^x+\cfrac{x(2-x)}{e^x}$