Property 20
Derivative of Indicator Function
\(f(x)=\operatorname{sgn}(g(x))\)
\(g(a)=0\qquad a\in\mathbb R\)
"a" point is the critical point of f function. As indicator function takes constant value its derivative is zero on the points where it is continuous. There is no derivative on the non-continuous points.
1.
\(f(x)=x^2\cdot\operatorname{sgn}(x-3)\)
\(\Rightarrow f'(4)=?\)
2.
\(f(x)=x^2+x\cdot\operatorname{sgn}(x+2)\)
\(\Rightarrow f'(3)=?\)
3.
\(f(x)=\operatorname{sgn}(x-3)+x^2\)
\(\Rightarrow f'(3)=?\)
4.
\(f(x)=|x-1|+\operatorname{sgn}(x-3)\cdot x\)
\(\Rightarrow f'(2)=?\)
5.
\(f(x)=x\cdot|x-2|+\operatorname{sgn}(|x-2|)\)
\(\Rightarrow f'(1)=?\)
6.
\(f(x)=|x^2-4x|+[\operatorname{sgn}(x-2)]\cdot x\)
\(\Rightarrow f'(3)=?\)
7.
\(f(x)=\begin{cases}|x^2-2x|&x>3\\x^2\cdot\operatorname{sgn}(x-2)&x\le3\end{cases}\)
\(\Rightarrow f'(2)=?\)
8.
\(f(x)=\operatorname{sgn}(x^2-5x+4)\cdot x+|x-2|\)
\(\Rightarrow f'(-3)=?\)
9.
\(f(x)=\operatorname{sgn}(x^2-9)\cdot x^3+x\)
\(\Rightarrow f'(2)=?\)
10.
\(f(x)=|x-3|+\operatorname{sgn}(x-5)\cdot\sqrt{x^2-1}+x\)
\(\Rightarrow f'(2)=?\)