Practice section 7 of 8

Differentiate the following expressions with respect to $x$, simplifying your answers whenever possible.

(a) $\tan^{-1}(e^{3x})$, [2]

(b) $5^{2x}$, [2]

(c) $\dfrac{ax}{\sqrt{x^3+1}}$, where $a$ is a constant. [3]

Differentiate the following with respect to $x$, leaving your answers in terms of $x$.

(a) $e^{ex}\cot\left(\dfrac x2\right)$, [3]

(b) $(2x)^{1/x}$. [4]

Differentiate the following with respect to $x$.

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

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

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

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

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