Review practice · Section 1 of 3

Differentiate each of the following with respect to x.

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

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

(iii) $\cos^4(2x)$2 marks

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

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

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

Differentiate the following with respect to $x$.

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

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

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 marks

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

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

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

(b) $5^{2x}$,2 marks

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

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

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

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