Question

Find the derivatives of the function:$f\left(x\right)=\left|3x+1\right|.$

Short answer

The derivative of$f\left(x\right)=\left|3x+1\right|is\frac{9x+3}{\left|3x+1\right|}.$

Step-by-step solution

Given information 

The given expression is$f\left(x\right)=\left|3x+1\right|.$

Simplification 

$$\begin{gathered} f\text{'}\left(x\right) \\ =\frac{d}{dx}\left(\left|3x+1\right|\right) \\ =\frac{d}{dx}\left(\sqrt{{\left(3x+1\right)}^2}\right) \\ =\frac{1}{2\sqrt{{(3x+1)}^2}}\times 2\left(3x+1\right)\times 3 \\ =\frac{9x+3}{\left|3x+1\right|} \\ Hence,f\text{'}\left(x\right)=\frac{9x+3}{\left|3x+1\right|}. \end{gathered}$$