1. **State the problem:** Differentiate the function $$f(x) = x^4 - x - 3 + \frac{1}{\sqrt{x}}$$. 2. **Rewrite the function:** Express the term $$\frac{1}{\sqrt{x}}$$ as a power of $$x$$: $$\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$$. So, $$f(x) = x^4 - x - 3 + x^{-\frac{1}{2}}$$. 3. **Differentiate each term separately:** - The derivative of $$x^4$$ is $$4x^3$$. - The derivative of $$-x$$ is $$-1$$. - The derivative of the constant $$-3$$ is $$0$$. - The derivative of $$x^{-\frac{1}{2}}$$ is $$-\frac{1}{2} x^{-\frac{3}{2}}$$ using the power rule. 4. **Combine the derivatives:** $$f'(x) = 4x^3 - 1 - \frac{1}{2} x^{-\frac{3}{2}}$$. 5. **Final answer:** $$\boxed{f'(x) = 4x^3 - 1 - \frac{1}{2} x^{-\frac{3}{2}}}$$