Question

Please help me do this mathematics question experts

Answer 1

`\textbf{(i)}\\\\~~~~~~~y= e^(3x) \sin 2x\\\\\\\implies (dy)/(dx) = (d)/(dx) \left( e^(3x) \sin 2x \right)\\\\\\~~~~~~~~~~~~=e^(3x) (d)/(dx) \sin 2x + \sin 2x (d)/(dx) e^(3x)\\\\\\~~~~~~~~~~~~=2e^(3x) \cos 2x + 3e^(3x)\sin 2x \\\\\\`

`\textbf{(ii)}\\\\~~~~~~~~y = (e^(3x))/(\cos 2x)\\\\\\\implies (dy)/(dx) = \frac d{dx} \left((e^(3x))/(\cos 2x) \right)\\\\\\~~~~~~~~~~~=(\cos 2x (d)/(dx)\left(e^(3x) \right) -e^(3x) (d)/(dx)( \cos 2x) )/(\cos^2 (2x))\\\\\\~~~~~~~~~~~=(3e^(3x) \cos 2x+2e^(3x)\sin2x )/(\cos^2(2x))\\\\\\~~~~~~~~~~~=(3e^(3x) \cos 2x)/( \cos^2(2x)) + (2e^(3x) \sin 2x)/(\cos^2(2x))\\\\\\~~~~~~~~~~~=3e^(3x)\sec (2x) + 2e^(3x)\sec (2x) \tan (2x)`

`\textbf{(iii)}\\\\~~~~~~~~~y=( 3x+2)^3\\\\\\\implies (dy)/(dx) = (d)/(dx)(3x+2)^3\\\\\\~~~~~~~~~~~=3(3x+2)^2 (d)/(dx) (3x+2)\\\\\\~~~~~~~~~~~=3(3x+2)^2 \cdot 3\\\\\\~~~~~~~~~~~=9(3x+2)^2`

`\textbf{(iv)}\\\\~~~~~~~~~x^2 +y^2 = 5xy\\\\\\\implies (d)/(dx)(x^2 +y^2) = 5(d)/(dx) (xy)\\\\\\\implies 2x + 2y (dy)/(dx) = 5\left(x (dy)/(dx) +y\right)\\ \\\\\implies 2x +2y (dy)/(dx) = 5x (dy)/(dx) +5y\\\\\\\implies 2y (dy)/(dx) - 5x (dy)/(dx) = 5y-2x\\\\\\\implies (2y-5x) (dy)/(dx) = 5y-2x\\\\\\\implies (dy)/(dx) = (5y-2x)/(2y-5x)`