Question

Can someone explain to me why e^x + xe^x +2x is the derivative of xe^x+x^2

Answer 1

that means (xe^x +x^2)'= e^x +xe^x +2x

Answer 2

Applying product rule for the first term at the left hand side gives us first 2 terms at the right hand side

Answer 3

what is the deriv. of just e^x? would it be xe^x

Answer 4

More generally:
\[\Large\bf\sf \left(a^x\right)'\quad=\quad a^x\ln a\]
Applying it to the exponential of base e,\[\Large\bf\sf \left(e^x\right)'\quad=\quad e^x\ln e\quad=\quad e^x\]

Answer 5

Did you understand the product rule part that OOPS pointed out?\[\Large\bf\sf (xe^x)'\quad=\quad \color{royalblue}{(x)'}e^x+x\color{royalblue}{(e^x)'}\]\[\Large\bf\sf (xe^x)'\quad=\quad \color{orangered}{(1)}e^x+x\color{orangered}{(e^x)}\]

Answer 6

yes i did! i understand it now. I just didnt realize that i needed to use the product rule. but i get it, thanks!