1. **Problem:** Find the derivative of the function $$f(x) = 3\sqrt{x} + 4x^6 + 5$$. 2. **Recall the formulas:** - The derivative of $$x^n$$ is $$nx^{n-1}$$. - The square root can be written as a power: $$\sqrt{x} = x^{\frac{1}{2}}$$. - The derivative of a constant is 0. 3. **Rewrite the function:** $$f(x) = 3x^{\frac{1}{2}} + 4x^6 + 5$$ 4. **Differentiate each term:** - $$\frac{d}{dx} \left(3x^{\frac{1}{2}}\right) = 3 \times \frac{1}{2} x^{\frac{1}{2} - 1} = \frac{3}{2} x^{-\frac{1}{2}}$$ - $$\frac{d}{dx} \left(4x^6\right) = 4 \times 6 x^{6-1} = 24x^5$$ - $$\frac{d}{dx} (5) = 0$$ 5. **Combine the results:** $$f'(x) = \frac{3}{2} x^{-\frac{1}{2}} + 24x^5$$ 6. **Rewrite the negative exponent:** $$f'(x) = \frac{3}{2\sqrt{x}} + 24x^5$$ **Final answer:** $$f'(x) = \frac{3}{2\sqrt{x}} + 24x^5$$