Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 327: 27

$$ f'(x)= 4e^x(1+ e^{x})^3.$$

Recall that $(e^x)'=e^x$ Recall that $(x^n)'=nx^{n-1}$ Since we have $$ f(x)=(1+ e^{x})^4$$ then the derivative $ f'(x)$, using the chain rule, is given by $$ f'(x)=4(1+ e^{x})^3(1+ e^{x})'=4e^x(1+ e^{x})^3.$$