7.1 Determine \( f^{\prime}(x) \) from first principles if \( f(x)=3 x^{2}-5 \) Determine \( \frac{d y}{d x} \) if: \( 7.2 .1 \quad y=2 x^{5}+\frac{4}{x^{3}} \) 7.2.2 \( \quad y=\left(\sqrt{x}-x^{2}\right)^{2} \) QUESTION 8 Determine \( f^{\prime}(x) \) from first principles if \( f(x)=-x^{2}+4 \) Determine the derivative of: 8.2.1 \( \quad y=3 x^{2}+10 x \) 8.2 \( \quad f(x)=\left(x-\frac{3}{2}\right. \)

To find \( f^{\prime}(x) \) from first principles for \( f(x)=3 x^{2}-5 \), you would use the definition of the derivative: \[ f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. \] Plugging in \( f(x+h) = 3(x+h)^{2} - 5 \) and simplifying gives you \( f^{\prime}(x) = 6x \). This means the slope of the parabola at any point \( x \) can be found with this neat formula! For \( y = 2x^{5} + \frac{4}{x^{3}} \), use the power rule for derivatives: \[ \frac{d y}{d x} = 10x^{4} - \frac{12}{x^{4}}. \] This showcases how polynomial forms react when you derive them! In a similar fashion, for \( y = \left(\sqrt{x} - x^{2}\right)^{2} \), applying the chain rule leads to \( \frac{d y}{d x} = 2(\sqrt{x} - x^{2})(\frac{1}{2\sqrt{x}} - 2x) \). Each derivative tells a story of growth rate and change at every point – pretty fascinating stuff!