Practice section 8 of 8

It is given that $y^x=e^y x^2$. Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$, simplifying your answer. [4]

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

(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]

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

A curve $C$ has equation $y^2=2\sin x+2xy$, where $0\leq x\leq2\pi$.

Show that $\dfrac{dy}{dx}=\dfrac{\cos x+y}{y-x}$. [2]