When Is A Function Differentiable?
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Given that \(f(x) = \left\{ {\begin{array}{*{20}{c}}{{x^2} + 3x + 2,\,\,\,x \le 0}\\{x + 2,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x > 0}\end{array}} \right.\)
Find \(f'(x)\) for the defined set of values.
\(f'(x) = \left\{ {\begin{array}{*{20}{c}}{2x + 3\,\,\,x \le 0}\\{1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x > 0}\end{array}} \right.\)
\begin{align}
&f(x)=\left\{\begin{array}{ll}
x^{2}+3 x+2 & x \leq 0 \\
x+2 & x>0
\end{array}\right. \\
&\text{At }x=0\;\; f(0)=2 \text{ for both.} \\
&\therefore f(x) \text { is not differentiable at } x=0 \\
&f^{\prime}(x)=\left\{\begin{array}{cl}
2 x+3 & x<0 \\
1 & x>0
\end{array}\right.
\end{align}
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