Use Implicit Differentiation to find $\frac{dy}{dx}$ for the following equations:

a$.\sqrt[\phantom{2}]{xy}=x^{2}y+1$

b$.ln(xy)+5x=7$

c$.3x^{3}y=6$

Find the derivative of the following functions:

a$.f(x)=\frac{co{s}^{-1}(x)}{x+1}$

b$.f(x)=x\sqrt[\phantom{2}]{4-x^2}+si{n}^{-1}(\frac{x}{3})$

Prove the following identity by using implicit differentiation and setting up an appropriate triangle:

$\frac{d}{dx}[ta{n}^{-1}(x)]=\frac{1}{1+x^2}$