a$)$ A function is given as $f(x)=\frac{1}{\sqrt[\phantom{2}]{2-x}}.$ By using first principle of derivative, find $f^{\text{'}}(x).$

b$)$ Find the first derivative of $f(x)=cot\sqrt[\phantom{2}]{4x^2+2}.$

c$)$ Consider the parametric equations of the curve

$x=sec(t^3),y=t^3+4.$

Obtain $\frac{dy}{dx}$ in terms of $t.$ Hence, find $\frac{dy}{dx}$ when $t=\frac{1}{2}.$

d$)$ Given that $e^y+xy+ln(1+2x)=1$ where $x0.$

i$.$ Show that $(e^y+x)\frac{d^{2}y}{d{x}^2}+e^y(\frac{dy}{dx}{)}^2+2\frac{dy}{dx}-\frac{4}{(1+2x{)}^2}=0.$

ii$.$ Hence, find the value of $\frac{dy}{dx}$ at the point