1. **State the problem:** Find the derivative of the function $$f(t) = \sqrt[5]{t} - \frac{1}{\sqrt[5]{t}}$$. 2. **Rewrite the function using exponents:** Recall that $$\sqrt[5]{t} = t^{\frac{1}{5}}$$ and $$\frac{1}{\sqrt[5]{t}} = t^{-\frac{1}{5}}$$. So, $$f(t) = t^{\frac{1}{5}} - t^{-\frac{1}{5}}$$. 3. **Recall the power rule for derivatives:** For any function $$g(t) = t^n$$, the derivative is $$g'(t) = n t^{n-1}$$. 4. **Apply the power rule to each term:** - Derivative of $$t^{\frac{1}{5}}$$ is $$\frac{1}{5} t^{\frac{1}{5} - 1} = \frac{1}{5} t^{-\frac{4}{5}}$$. - Derivative of $$- t^{-\frac{1}{5}}$$ is $$- \left(-\frac{1}{5} t^{-\frac{1}{5} - 1}\right) = \frac{1}{5} t^{-\frac{6}{5}}$$. 5. **Combine the derivatives:** $$f'(t) = \frac{1}{5} t^{-\frac{4}{5}} + \frac{1}{5} t^{-\frac{6}{5}} = \frac{1}{5} \left(t^{-\frac{4}{5}} + t^{-\frac{6}{5}}\right)$$. 6. **Final answer:** $$f'(t) = \frac{1}{5} \left(t^{-\frac{4}{5}} + t^{-\frac{6}{5}}\right)$$. This derivative tells us the rate of change of the function $$f(t)$$ with respect to $$t$$.