Property 21
Derivative of Greatest Integer Function
\(f(x)=\left[\!\left[ g(x)\right]\!\right]\)
\(g(a)\in\mathbb Z\qquad a\in\mathbb R\)
"a" point is the critical point of f function. As greatest integer 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.
\(f(x)=\left[\!\left[ x\right]\!\right]\)
\(\Rightarrow f'(4)=?\)
2.
\(f(x)=\left[\!\left[ x\right]\!\right]\)
\(\Rightarrow f'\left(\samefrac{3}{2}\right)=?\)
3.
\(f(x)=\left[\!\left[ x-2\right]\!\right]\)
\(\Rightarrow f'\left(\samefrac{1}{2}\right)=?\)
4.
\(f(x)=(x^2+1)\cdot\left[\!\left[ x+3\right]\!\right]\)
\(\Rightarrow f'\left(\samefrac{3}{2}\right)=?\)
5.
\(f(x)=x^3\cdot\left[\!\left[ \samefrac x3+2\right]\!\right]\)
\(\Rightarrow f'(2)=?\)
6.
\(f(x)=\left[\!\left[ \samefrac{x+5}2\right]\!\right]\cdot\operatorname{sgn}(x-3)\)
\(\Rightarrow f'(4)=?\)
7.
\(f(x)=3\cdot\left[\!\left[ x^2-1\right]\!\right]\cdot(x^2+1)\)
\(\Rightarrow f'\left(\samefrac{3}{2}\right)=?\)
8.
\(f(x)=\left[\!\left[ \samefrac{x-2}7\right]\!\right]\cdot x^5\)
\(\Rightarrow f'(9)=?\)
9.
\(f(x)=3\cdot\left[\!\left[ (x-1)^2\right]\!\right]\)
\(\Rightarrow f'(1)=?\)
10.
\(f(x)=(x^2-16)\cdot\left[\!\left[ \samefrac x4-1\right]\!\right]\)
\(\Rightarrow f'(2)=?\)
11.
\(f(x)=|x^2-4|+\operatorname{sgn}(x-3)+\left[\!\left[ x+\samefrac{3}{2}\right]\!\right]\)
\(\Rightarrow f'(4)=?\)