Property 19
Derivative of Absolute Value Function
\(f(x)=|g(x)|\)
\(g(a)=0\qquad a\in\mathbb R\)
"a" point is the critical point of the function
\(f(x)=|mx+n|+|px+r|\)
f function is continuous in all real numbers. However, as right-hand and left-hand derivatives are different there is no derivative on their breaking points.
Breaking points are; \(-\samefrac nm\) and \(-\samefrac rp\) dir.
1.
\(f(x)=|x-3|\)
\(\Rightarrow f'(2)=?\)
2.
\(f(x)=|3x-20|\)
\(\Rightarrow f'(7)=?\)
3.
\(f(x)=|x^2-4x|\)
\(\Rightarrow f'(2)=?\)
4.
\(f(x)=|x-2|\)
\(\Rightarrow f'(2)=?\)
5.
\(f(x)=|\cos x-4|\)
\(\Rightarrow f'\left(\samefrac{\pi}{2}\right)=?\)
6.
\(f(x)=|x^2-3x-4|\)
\(\Rightarrow f'(2)=?\)
7.
\(f(x)=x^2\cdot|x|\)
\(\Rightarrow f'(0)=?\)
8.
\(f(x)=x^2\cdot|3x+15|\)
\(\Rightarrow f'(2)=?\)
9.
\(f(x)=x^3\cdot|2x-8|\)
\(\Rightarrow f'(1)=?\)
10.
\(f(x)=|x^2-4x+4|\)
\(\Rightarrow f'(2)=?\)