Exercise · ZW-DERIVATIVE-016 · ★★☆ Intermediate
For \(f(x)=|x|\), what happens with \(f'(0)\)?
Choose one option.
Show answer
- C. It does not exist: the one-sided limits of the quotient are \(1\) and \(-1\).
Explanation
The quotient is \(|h|/h\), equal to \(1\) for \(h>0\) and \(-1\) for \(h<0\). The one-sided limits differ so the limit does not exist: the graph has a corner.
Review the concept: Derivative
The derivative of \(f\) at \(a\) is the limit of the difference quotient as \(h\to0\). It measures the instantaneous rate of change and is the slope of the tangent line at \((a,f(a))\).
\[f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]
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