Compare direct and logarithmic implicit differentiation

(a) Differentiate $\dfrac{x-2x^3}{\ln x}$ with respect to $x$.2 marks

(b) Given that $0<x<\dfrac\pi2$, show that $\dfrac d{dx}[\sin^{-1}(\cos x)]=k$, where $k$ is a real constant to be determined.3 marks

(c) Given that $e^{xy}=(1+y^2)^2$, find $\dfrac{dy}{dx}$ in terms of $x$ and $y$, simplifying your answer.4 marks