Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 328: 92

See explanation

Compute the derivative of $g(x)e^{-x}$: Using the product rule for the derivatives it follows: $$(g(x)e^{-x})'=g'(x)e^{-x}+g(x)(e^{-x})'$$ $$(g(x)e^{-x})'=g'(x)e^{-x}-g(x)e^{-x}$$ $$(g(x)e^{-x})'=g(x)e^{-x}-g(x)e^{-x}$$ $$(g(x)e^{-x})'=0$$ $$g(x)e^{-x}=C$$ $$g(x)e^{-x}e^{x}=Ce^{x}$$ $$g(x)e^{-x+x}=Ce^{x}$$ $$g(x)e^{0}=Ce^{x}$$ $$g(x)=Ce^{x}$$