1. **Problem:** Find the derivative functions $f'$ for the given functions. 2. **Recall the derivative rules:** - Power rule: $\frac{d}{dx} x^n = n x^{n-1}$ - Derivative of a constant: $\frac{d}{dx} c = 0$ - Derivative of $\frac{1}{x} = x^{-1}$ is $-x^{-2} = -\frac{1}{x^2}$ - Derivative of $\sqrt{x} = x^{1/2}$ is $\frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}$ 3. **Calculate each derivative:** **a)** $f(x) = x^3$ $$f'(x) = 3x^{3-1} = 3x^2$$ **b)** $f(x) = 1$ $$f'(x) = 0$$ **c)** $f(x) = \frac{1}{x} = x^{-1}$ $$f'(x) = -1 \cdot x^{-1-1} = -x^{-2} = -\frac{1}{x^2}$$ **e)** $f(x) = x^5$ $$f'(x) = 5x^{5-1} = 5x^4$$ **f)** $f(x) = \sqrt{x} = x^{1/2}$ $$f'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}$$ **h)** $f(x) = x$ $$f'(x) = 1$$ **Final answers:** - a) $f'(x) = 3x^2$ - b) $f'(x) = 0$ - c) $f'(x) = -\frac{1}{x^2}$ - e) $f'(x) = 5x^4$ - f) $f'(x) = \frac{1}{2\sqrt{x}}$ - h) $f'(x) = 1$