Exercise · ZW-DERIVATIVE-005 · ★★☆ Intermediate
Using the limit definition of the derivative, what is \(f'(2)\) if \(f(x)=x^{3}\)?
Choose one option.
Show answer
- A. \(12\)
Explanation
The difference quotient is \(\dfrac{f(2+h)-f(2)}{h}=\frac{\left(h + 2\right)^{3} - 8}{h}\). As h → 0 it tends to \(12\), which is f'(2).
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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