Question

True or False The derivative of \(y=\sqrt[3]{x}\) is \(\frac{1}{3 x^{2 / 3}} .\) Justify your answer.

Short answer

True, the derivative of \(y=\sqrt[3]{x}\) is indeed \(\frac{1}{3x^{\frac{2}{3}}}\)

Step-by-step solution

Identify the Power in the Original Function

Rewrite the given function \(y=\sqrt[3]{x}\) in a form that shows the exponent explicitly. This results in \(y = x^{\frac{1}{3}}\). The power of \(x\) here is \(\frac{1}{3}\).

Apply the Power Rule

By the power rule, the derivative of \(x^n\) is \(nx^{n-1}\). Substituting \(\frac{1}{3}\) for \(n\), the derivative is \(\frac{1}{3}x^{(\frac{1}{3}-1)}\).

Simplify the Derivative Formula

Simplify the derivative, \(\frac{1}{3}x^{(\frac{1}{3}-1)}\), to the result \(\frac{1}{3}x^{-\frac{2}{3}}\). Recall that \(x^{-n} = \frac{1}{x^n}\), applying this rule the derivative simplifies to \(\frac{1}{3x^{\frac{2}{3}}}\).