1. **State the problem:** Differentiate the function $$f(x) = 4x^4 + 2x^3 + \sqrt{x} + 20$$ with respect to $$x$$. 2. **Recall the differentiation rules:** - Power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ for any real number $$n$$. - Constant rule: derivative of a constant is 0. - For $$\sqrt{x}$$, rewrite as $$x^{1/2}$$ and apply the power rule. 3. **Apply the power rule to each term:** - $$\frac{d}{dx} 4x^4 = 4 \times 4 x^{4-1} = 16x^3$$ - $$\frac{d}{dx} 2x^3 = 2 \times 3 x^{3-1} = 6x^2$$ - $$\frac{d}{dx} x^{1/2} = \frac{1}{2} x^{1/2 - 1} = \frac{1}{2} x^{-1/2}$$ - $$\frac{d}{dx} 20 = 0$$ 4. **Combine all derivatives:** $$f'(x) = 16x^3 + 6x^2 + \frac{1}{2} x^{-1/2} + 0 = 16x^3 + 6x^2 + \frac{1}{2} x^{-1/2}$$ 5. **Compare with given options:** - Option a: $$16x^3 + 6x^2 + 12x^{1/2}$$ (incorrect coefficient for $$x^{1/2}$$ term) - Option b: $$16x^3 + 6x^2 + x^{1/2}$$ (incorrect coefficient for $$x^{1/2}$$ term) - Option c: $$12x^3 + 6x^2 + 12x^{-1/2}$$ (incorrect coefficients) - Option d: $$4x^3 + 2x^2 + x^{-1/2}$$ (incorrect coefficients) None of the options match the correct derivative. **Final answer:** e. None of the given options is correct.