Exercise · ZW-DERIVATIVE-015 · ★☆☆ Basic
For \(f(x)=x^3\) at \(c=1\), what does the difference quotient \(\dfrac{f(1+h)-f(1)}{h}\) equal for \(h\ne0\)?
Choose one option.
Show answer
- D. \(3+3h+h^2\)
Explanation
\((1+h)^3-1=3h+3h^2+h^3\); dividing by \(h\) gives \(3+3h+h^2\), which tends to \(f'(1)=3\).
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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