Exercise · ZW-DERIVATIVE-002 · ★★☆ Intermediate
If a function is differentiable at a, then...
Choose one option.
Show answer
- D. it is continuous at a
Explanation
Every function differentiable at a is continuous at a. The converse is false: |x| is continuous at 0 but not differentiable there.
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-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?
- ZW-DERIVATIVE-009 Let f(x)=|x|. What happens with f'(0)?