Since the function [i]f[/i] is always increasing, the slope of its tangent line always remains positive. [color=#0000ff]Thus, the output of its derivative function, f', is always positive. [/color]

[math]x=e^y[/math] or [math]e^y=x[/math]

[math]\frac{d}{dx}\left(e^y\right)=\frac{d}{dx}\left(x\right)[/math][br][br][math]e^y\cdot\frac{dy}{dx}=1[/math][br][br][math]\frac{dy}{dx}=\frac{1}{e^y}[/math]

[math]\frac{dy}{dx}=\frac{1}{e^y}=\frac{1}{e^{ln\left(x\right)}}=\frac{1}{x}[/math]. [br][br]Thus, [math]\frac{d}{dx}\left(ln\left(x\right)\right)=\frac{1}{x}[/math]. [br][br][color=#0000ff]Thus, the graph of the blue function is the right branch of the function[/color] [math]y=\frac{1}{x}[/math]. [color=#0000ff]Note that it is the right branch only because the domain of the function [/color][math]f\left(x\right)=ln\left(x\right)[/math][color=#0000ff] is [/color][math]\left(0,\infty\right)[/math][color=#0000ff].[/color]