Question

If $f\left(x\right)=\sqrt{x^2+6x+9},\mathrm{then}f\text{'}\left(x\right)$ is equal to
(a) $1\mathrm{for}x<-3$
(b) $-1\mathrm{for}x<-3$
(c) $1\mathrm{for}\mathrm{all}x\in R$
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{b}\right)-1\mathrm{for}x<-3 \\ \mathrm{We}\mathrm{have},f\left(x\right)=\sqrt{x^2+6x+9} \\ =\sqrt{{\left(x+3\right)}^2} \\ =\left|x+3\right| \\ f\left(x\right)=\left\{\begin{array}{l}x+3,x\ge -3\\ -x-3,x<-3\end{array}\right. \\ \Rightarrow f\text{'}\left(x\right)=\left\{\begin{array}{l}1,x\ge -3\\ -1,x<-3\end{array}\right. \\ \therefore f\text{'}\left(x\right)=-1\mathrm{for}x<-3 \end{gathered}$$