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Question
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
Options
\[1 \text { for x } < - 3\]
\[- 1\text { for x} < - 3\]
\[1\text { for all } x \in R\]
none of these
Solution
\[- 1\text { for x} < - 3 \]
\[\text { We have, }f\left( x \right) = \sqrt{x^2 + 6x + 9}\]
\[ = \sqrt{\left( x + 3 \right)^2} \]
\[ = \left| x + 3 \right| \]
`f(x) ={(x+3, x>=-3) ,(-x-3, x<-3):}`
`rArrf'(x)={[1, x>=-3], [-1, x<-3]:}`
`thereforef'(x)=-1` for `x<-3`
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