Derivative of a fractional power function evaluated at a point

Let $f(x) = x^{1/5}$ . Use the power rule to find $f'(x)$ , and then evaluate $f'(32)$ .

Hint

Apply the power rule $\frac{d}{dx}[x^n] = n x^{n-1}$ with $n = \frac{1}{5}$ . Then substitute $x = 32$ , remembering that $32 = 2^5$ .

Step-by-step solution

By the power rule, $f'(x) = \frac{1}{5} x^{1/5 - 1} = \frac{1}{5} x^{-4/5}$ . This can also be written as $f'(x) = \dfrac{1}{5x^{4/5}}$ .

Now substitute $x = 32$ . Since $32 = 2^5$ , we have $32^{4/5} = (2^5)^{4/5} = 2^4 = 16$ . So $f'(32) = \dfrac{1}{5 \cdot 16} = \dfrac{1}{80}$ .

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