Review practice · Section 2 of 3

Differentiate the following with respect to $x$.

(i) $\cos^{-1}(\sin x)$, where $\dfrac\pi2<x<\dfrac{3\pi}2$,2 marks

(ii) $\ln\sqrt{\dfrac{e^x+1}{1-e^{-x}}}$.3 marks

Given that $y^3+e^{\tan y}=\cos\left(2x+\dfrac\pi3\right)$, find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.4 marks

The variables $x$ and $y$ are related by $3^y=xy-\sec x$. Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.4 marks

Given that $y=(2x+1)^x$ for $x>0$, find $\dfrac{dy}{dx}$.3 marks

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

(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