Exercise · ZW-DERIVATIVE-019 · ★★★ Advanced
Let \(f(x)=x^{1/3}\) (real cube root). What happens with \(f'(0)\)?
Choose one option.
Show answer
- C. It does not exist: the quotient \(h^{-2/3}\) grows without bound.
Explanation
The difference quotient is \(h^{1/3}/h=h^{-2/3}\to+\infty\). The tangent line is vertical, but the derivative, as a real number, does not exist. The cube root is defined for \(x<0\).
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 \((a,f(a))\).
\[f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]
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