Practice section 6 of 8

Find $\dfrac d{dx}[\sin^{-1}\sqrt{1-x^2}]$.

(a) Differentiate the following with respect to $x$, giving your answers as single fractions.

(i) $\ln\left(\dfrac{x}{\sqrt{1-2x}}\right)$,

(ii) $\dfrac1{\cos^{-1}(x^2)}$.

(b) The variables $x$ and $y$ are related by $e^{xy^2}=y(x^2+2e^x)$.

Find the value of $\dfrac{dy}{dx}$ when $x=0$ and $y=\dfrac12$.

Differentiate each of the following with respect to x.

(i) $e^{\sec x}$ [1]

(ii) $\tan^{-1}(x^2)$ [2]

(iii) $\cos^4(2x)$ [2]

Differentiate the following expressions with respect to $x$, simplifying your answers as far as possible:

(a) $\tan^{-1}\left(\dfrac2x\right)$, [3]

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

Differentiate the following with respect to $x$.

(i) $\sqrt{\cos^{-1}\left(\dfrac x2\right)}$, [2]

(ii) $\ln\sqrt{\dfrac{(x+1)^3}{x^2-1}}$. [2]

Differentiate with respect to $x$, giving your answers as a single fraction,

(a) $\ln\sqrt{a^2-x^2}$, where $a$ is a constant, [2]

(b) $\tan^{-1}\left(\dfrac1{2x}\right)$. [2]