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a)
f\left(x\right)=\sqrt{x^2+1}{,}\ x\in\mathbb{R}
\sqrt{x^2+1}=\sqrt{x^2}+\sqrt{1}=x+1
f'\left(x\right)=\frac{x}{\sqrt{x^2+1}}
b)
f'\left(x\right)=\frac{x}{\sqrt{x^2+1}}
f'\left(0\right)=\frac{0}{\sqrt{0^2+1}}=\frac{0}{\left|1\right|}=0
c)
f\left(x\right)=0
x=0