Exercise · ZW-DERIVATIVE-009 · ★★★ Advanced
Let \(f(x)=|x|\). What happens with \(f'(0)\)?
Choose one option.
Show answer
- A. It does not exist: the one-sided limits of the difference quotient are \(1\) and \(-1\)
Explanation
For \(h>0\), \(\frac{|h|}{h}=1\); for \(h<0\), \(\frac{|h|}{h}=-1\). The one-sided limits differ, so \(f'(0)\) does not exist, even though \(f\) is continuous at 0.
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 \((a,f(a))\).
\[f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]
More practice problems: derivative
- ZW-DERIVATIVE-001 The number f'(a) represents...
- ZW-DERIVATIVE-002 If a function is differentiable at a, then...
- ZW-DERIVATIVE-003 Using the limit definition of the derivative, what is f'(2) if f(x)=x^2 + x?
- ZW-DERIVATIVE-004 Using the limit definition of the derivative, what is f'(1) if f(x)=3 x^2 - 2 x?
- ZW-DERIVATIVE-005 Using the limit definition of the derivative, what is f'(2) if f(x)=x^3?
- ZW-DERIVATIVE-006 Using the limit definition of the derivative, what is f'(4) if f(x)=√x?
- ZW-DERIVATIVE-007 Using the limit definition of the derivative, what is f'(2) if f(x)=(1)/(x)?
- ZW-DERIVATIVE-008 Which limit defines the derivative of f at a?