1. Problem: Find the derivatives of the given functions. 2. Recall the power rule for derivatives: $$\frac{d}{dx} x^n = n x^{n-1}$$ and the derivative of a constant is 0. 3. a) For $$f(x) = x^6 + 2x - 1$$ $$f'(x) = \frac{d}{dx} x^6 + \frac{d}{dx} 2x - \frac{d}{dx} 1 = 6x^5 + 2 - 0 = 6x^5 + 2$$ 4. b) For $$h(x) = \frac{3}{5} x^5 - 4x^{-2}$$ $$h'(x) = \frac{3}{5} \cdot 5 x^{4} - 4 \cdot (-2) x^{-3} = 3 x^{4} + 8 x^{-3}$$ 5. c) For $$f(t) = \sqrt{t} + \frac{1}{t^3} = t^{\frac{1}{2}} + t^{-3}$$ $$f'(t) = \frac{1}{2} t^{-\frac{1}{2}} - 3 t^{-4} = \frac{1}{2} t^{-\frac{1}{2}} - 3 t^{-4}$$ 6. d) For $$f(x) = x^{x+1} + \frac{1}{x}$$ Use logarithmic differentiation for $$x^{x+1}$$: Let $$y = x^{x+1}$$, then $$\ln y = (x+1) \ln x$$ Differentiate both sides: $$\frac{1}{y} y' = \ln x + (x+1) \frac{1}{x} = \ln x + 1 + \frac{1}{x}$$ Multiply both sides by $$y$$: $$y' = x^{x+1} \left( \ln x + 1 + \frac{1}{x} \right)$$ Derivative of $$\frac{1}{x}$$ is $$-x^{-2}$$. So, $$f'(x) = x^{x+1} \left( \ln x + 1 + \frac{1}{x} \right) - x^{-2}$$