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Please solve RD Sharma class 12 chapter 10 Differentiation exercise Multiple choice question 21 maths textbook solution

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Answer:

        -1 \text { for } x<-3

Hint:

        Differentiate the function w.r.t x

Given:

        f(x)=\sqrt{x^{2}+6 x+9}, f^{\prime}(x)

Solution:  

        \begin{aligned} &f(x)=\sqrt{x^{2}+6 x+9} \\\\ &x^{2}+6 x+9=(x+3)(x+3)=(x+3)^{2} \end{aligned}

        \begin{aligned} &f(x)=\sqrt{(x+3)^{2}}=|x+3| \\\\ &f(x)=\left\{\begin{array}{cc} x+3 & x \geq-3 \\\\ -x-3 & x<-3 \end{array}\right. \end{aligned}

        \begin{aligned} &f^{\prime}(x)=\left\{\begin{array}{cl} 1 & x \geq-3 \\\\ -1 & x<-3 \end{array}\right. \\\\ &f^{\prime}(x)=-1 \text { For } x<-3 \end{aligned}

 

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