a$)$ Find the derivative $\frac{dy}{dx}$ for $y=\frac{e^{x^2}+lnx}{sin(3x+2)}.$

b$)$ If $y=x^{2}cos3x,$ show that

$x^2\frac{d^{2}y}{d{x}^2}-4x\frac{dy}{dx}+(9x^2+6)y=0.$

c$)$ The parametric equations of a curve are given by;

$x=t-\frac{2}{t},$ and $,y=\frac{\sqrt[\phantom{2}]{t^2+2}}{2t},$ where $t0$

i$.,$ Show that $\frac{dy}{dx}=-(t^2+2{)}^{-\frac{3}{2}}.$

ii$.$ Evaluate the value of $\frac{d^{2}y}{d{x}^2}$ when $t=1.$

Question $2$:

a$)$ Find the derivative with respect to $x$ of $f(x)=\sqrt[\phantom{2}]{3x-2}$ using the definition of derivative.

b$)$ If $y=\frac{1}{\sqrt[\phantom{2}]{2x^2+1}},$ show that $y^{\text{'}\text{'}}-y^{\text{'}}+2y^5=\frac{8x^2}{(\sqrt[\phantom{2}]{2x^2+1}{)}^5}+\frac{2x}{(\sqrt[\phantom{2}]{2x^2+1}{)}^3}$

c$)$ Using implicit differentiation, find $\frac{dy}{dx}$ for $\sqrt[\phantom{2}]{x}y+cos{y}^2=e^{4x}$