[tex]\textbf{(i)}\\\\~~~~~~~y= e^{3x} \sin 2x\\\\\\\implies \dfrac{dy}{dx} = \dfrac{d}{dx} \left( e^{3x} \sin 2x \right)\\\\\\~~~~~~~~~~~~=e^{3x} \dfrac{d}{dx} \sin 2x + \sin 2x \dfrac{d}{dx} e^{3x}\\\\\\~~~~~~~~~~~~=2e^{3x} \cos 2x + 3e^{3x}\sin 2x \\\\\\[/tex]
[tex]\textbf{(ii)}\\\\~~~~~~~~y = \dfrac{e^{3x}}{\cos 2x}\\\\\\\implies \dfrac{dy}{dx} = \dfrac d{dx} \left(\dfrac{e^{3x}}{\cos 2x} \right)\\\\\\~~~~~~~~~~~=\dfrac{\cos 2x \dfrac{d}{dx}\left(e^{3x} \right) -e^{3x} \dfrac{d}{dx}( \cos 2x) }{\cos^2 (2x)}\\\\\\~~~~~~~~~~~=\dfrac{3e^{3x} \cos 2x+2e^{3x}\sin2x }{\cos^2(2x)}\\\\\\~~~~~~~~~~~=\dfrac{3e^{3x} \cos 2x}{ \cos^2(2x)} + \dfrac{2e^{3x} \sin 2x}{\cos^2(2x)}\\\\\\~~~~~~~~~~~=3e^{3x}\sec (2x) + 2e^{3x}\sec (2x) \tan (2x)[/tex]
[tex]\textbf{(iii)}\\\\~~~~~~~~~y=( 3x+2)^3\\\\\\\implies \dfrac{dy}{dx} = \dfrac{d}{dx}(3x+2)^3\\\\\\~~~~~~~~~~~=3(3x+2)^2 \dfrac{d}{dx} (3x+2)\\\\\\~~~~~~~~~~~=3(3x+2)^2 \cdot 3\\\\\\~~~~~~~~~~~=9(3x+2)^2[/tex]
[tex]\textbf{(iv)}\\\\~~~~~~~~~x^2 +y^2 = 5xy\\\\\\\implies \dfrac{d}{dx}(x^2 +y^2) = 5\dfrac{d}{dx} (xy)\\\\\\\implies 2x + 2y \dfrac{dy}{dx} = 5\left(x \dfrac{dy}{dx} +y\right)\\ \\\\\implies 2x +2y \dfrac{dy}{dx} = 5x \dfrac{dy}{dx} +5y\\\\\\\implies 2y \dfrac{dy}{dx} - 5x \dfrac{dy}{dx} = 5y-2x\\\\\\\implies (2y-5x) \dfrac{dy}{dx} = 5y-2x\\\\\\\implies \dfrac{dy}{dx} = \dfrac{5y-2x}{2y-5x}[/tex]
