Derivative of a fractional power at a point

Let $f(x) = \sqrt[3]{x} = x^{1/3}$ .

Find $f'(1)$ for $f(x) = x^{1/3}$ .

Hint

Apply the power rule $\frac{d}{dx}[x^n] = n x^{n-1}$ with $n = 1/3$ , then substitute $x = 1$ .

Step-by-step solution

Using the power rule with $n = 1/3$ , we get $f'(x) = \frac{1}{3}x^{1/3 - 1} = \frac{1}{3}x^{-2/3}$ . This can also be written as $f'(x) = \dfrac{1}{3x^{2/3}}$ .

Now substitute $x = 1$ : since $1^{-2/3} = 1$ , we have $f'(1) = \frac{1}{3}\cdot 1 = \frac{1}{3}$ .

Derivative of a fractional power at a point

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