Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 327: 16

$$ y=4x+1.$$

We have $$ y'=4e^{4x}\Longrightarrow y'(0)=4.$$ Then the slope of the tangent line at $ x=0$ is $ m= 4$. Hence the equation of the tangent line is given by $$ y=mx+c=4x+c.$$ Since $ y(0)=1$, then $ c=1$, hence the tangent line at $ x=0$ is given by $$ y=4x+1.$$