1. **Problem 4.2:** Determine the equation of the tangent line to the curve $f(x) = 5x^2 + 4x - 1$ at $x = 3$. 2. **Find the derivative $f'(x)$:** $$f'(x) = \frac{d}{dx}(5x^2 + 4x - 1) = 10x + 4$$ 3. **Evaluate the slope of the tangent line at $x=3$:** $$f'(3) = 10(3) + 4 = 30 + 4 = 34$$ 4. **Find the point on the curve at $x=3$:** $$f(3) = 5(3)^2 + 4(3) - 1 = 5(9) + 12 - 1 = 45 + 12 - 1 = 56$$ 5. **Use point-slope form of the line:** $$y - y_1 = m(x - x_1)$$ where $m=34$, $x_1=3$, and $y_1=56$. 6. **Write the equation of the tangent line:** $$y - 56 = 34(x - 3)$$ $$y = 34x - 102 + 56 = 34x - 46$$ --- 7. **Problem 4.4:** Find the derivative $\frac{dy}{dx}$ of $$y = \frac{\frac{7}{x} - 5x + 3x^4}{\sqrt{x}}$$ 8. **Rewrite $y$ for easier differentiation:** $$y = \left(7x^{-1} - 5x + 3x^4\right) x^{-\frac{1}{2}} = 7x^{-1} x^{-\frac{1}{2}} - 5x x^{-\frac{1}{2}} + 3x^4 x^{-\frac{1}{2}}$$ $$= 7x^{-\frac{3}{2}} - 5x^{\frac{1}{2}} + 3x^{\frac{7}{2}}$$ 9. **Differentiate term-by-term using power rule $\frac{d}{dx} x^n = n x^{n-1}$:** $$\frac{dy}{dx} = 7 \cdot \left(-\frac{3}{2}\right) x^{-\frac{5}{2}} - 5 \cdot \frac{1}{2} x^{-\frac{1}{2}} + 3 \cdot \frac{7}{2} x^{\frac{5}{2}}$$ $$= -\frac{21}{2} x^{-\frac{5}{2}} - \frac{5}{2} x^{-\frac{1}{2}} + \frac{21}{2} x^{\frac{5}{2}}$$ 10. **Final derivative:** $$\boxed{\frac{dy}{dx} = -\frac{21}{2} x^{-\frac{5}{2}} - \frac{5}{2} x^{-\frac{1}{2}} + \frac{21}{2} x^{\frac{5}{2}}}$$