$\def\D{\displaystyle}\def\frac{\dfrac}$
1 (CIE 2012, s, paper 12, question 5)
Differentiate the following with respect to $\D x.$
(i) $\D (2-x^2)\ln (3x+1)$ [3]
(ii) $\D \frac{4-\tan 2x}{5x}$ [3]
2 (CIE 2012, s, paper 21, question 7)
Given that $\D f(x)=x^2-\frac{648}{\sqrt{x}},$ find the value of $\D x$ for which $\D f''(x)=0.$ [6]
3 (CIE 2012, w, paper 11, question 5)
Given that $\D y=\frac{x^2}{\cos 4x},$ find
(i) $\D \frac{dy}{dx}$, [3]
(ii) the approximate change in $\D y$ when $\D x$ increases from $\D \frac{\pi}{4}$to $\D \frac{\pi}{4}+p$, where $\D p$ is small. [2]
4 (CIE 2013, s, paper 22, question 7)
Differentiate with respect to $\D x$,
(i) $\D (3-5x)^{12}$ [2]
(ii)$\D x^2\sin x$ [2]
(iii)$\D \frac{\tan x}{1+e^{2x}}$ [4]
5 (CIE 2014, s, paper 21, question 7)
Given that a curve has equation $\displaystyle y=\frac{1}{x}+2\sqrt{x},$ where $\D x>0,$ find
(i) $\D \frac{dy}{dx},$ [2]
(ii) $\D \frac{d^2y}{dx^2}.$ [2]
Hence, or otherwise, find
(iii) the coordinates and nature of the stationary point of the curve. [4]
6 (CIE 2014, s, paper 21, question 10)
Find $\D \frac{dy}{dx}$ when
(i) $\D y=\cos 2x \sin\left(\frac{x}{3}\right),$ [4]
(ii) $\D y=\frac{\tan x}{1+\ln x}.$ [4]
7 (CIE 2014, s, paper 22, question 7)
Differentiate with respect to $\D x$
(i) $\D x^4e^{3x},$ [2]
(ii) $\D \ln(2+\cos x),$ [2]
(iii) $\D \frac{\sin x}{1+\sqrt{x}}.$ [3]
8 (CIE 2015, w, paper 23, question 3)
(a) Given that $\D y=\frac{x^3}{2-x^2},$ find $\D \frac{dy}{dx}.$ [3]
(b) Given that $\D y=x\sqrt{4x+6}, $ show that $\D \frac{dy}{dx}= \frac{k(x+1)}{\sqrt{4x+6}}$ and state the value of $\D k.$ [3]
9 (CIE 2016, march, paper 22, question 1)
Two variables $\D x$ and $\D y$ are such that $\D y=\frac{5}{\sqrt{x-9}}$ for $\D x>9$.
(i) Find an expression for $\D \frac{dy}{dx}.$ [2]
(ii) Hence, find the approximate change in $\D y$ as $\D x$ increases from 13 to $\D 13+h$, where $\D h$ is small. [2]
10 (CIE 2016, s, paper 12, question 6)
Show that $\D \frac{d}{dx}(e^{3x} \sqrt{4x+1})$ can be written in the form $\D \frac{e^{3x}(px+q)}{\sqrt{4x+1}},$ where $p$ and $q$ are integers to be found. [5]
11 (CIE 2017, s, paper 11, question 10)
(a) Given that $\D y=\frac{e^{3x}}{4x^2+1},$ find $\D \frac{dy}{dx}.$ [3]
(b) Variables $x,y,$ and $t$ are such that $\D y=4\cos \left(x+\frac{\pi}{3}\right)+ 2\sqrt{3}\sin\left(x+\frac{\pi}{3}\right)$ and $\D \frac{dy}{dt}=10.$
(i) Find the value of $\D \frac{dy}{dx}$ when $\D x=\frac{\pi}{2}.$ [3]
(ii) Find the value of $\D\frac{dx}{dt}$ when $\D x=\frac{\pi}{2}.$ [2]
12 (CIE 2017, s, paper 12, question 2)
It is given that $\D \frac{(5x^2+4)^{\frac{1}{2}}}{x+1}.$ Showing all your working, find the exact value of $\D \frac{dy}{dx}$ when $\D x = 3.$ [5]
13 (CIE 2017, s, paper 21, question 3)
The variables $\D x$ and $\D y$ are such that $\D y = \ln(x^2 + 1).$
(i) Find an expression for $\D \frac{dy}{dx}.$ [2]
(ii) Hence, find the approximate change in $\D y$ when $\D x $ increases from 3 to $\D 3 + h,$ where $h$ is small. [2]
14 (CIE 2017, s, paper 23, question 7)
Differentiate with respect to $\D x$.
(i) $\D (1+4x)^{10}\cos x,$
(ii) $\D\frac{e^{4x-5}}{\tan x}.$
15 (CIE 2017, w, paper 12, question 4)
Given that $\D y= \frac{\ln(3x^2+2)}{x^2+1},$ find the value of $\D \frac{dy}{dx}$ when $\D x = 2,$ giving your answer as $\D a + b \ln 14,$ where $a$ and $b$ are fractions in their simplest form. [6]
16 (CIE 2017, w, paper 21, question 7)
Find $y$ in terms of $x$, given that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=6 x+\frac{2}{x^{3}}$ and that when $x=1, y=3$ and $\frac{\mathrm{d} y}{\mathrm{~d} x}=1$.
17 (CIE 2017, w, paper 21, question 12)
(i) Differentiate $(\cos x)^{-1}$ with respect to $x$.
(ii) Hence find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ given that $y=\tan x+4(\cos x)^{-1}$.
(iii) Using your answer to part (ii) find the values of $x$ in the range $0 \leqslant x \leqslant 2 \pi$ such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=4$. [6]
18 (CIE 2017, w, paper 22, question 10)
(i) Without using a calculator, solve the equation $6 c^{3}-7 c^{2}+1=0$. $[5]$ It is given that $y=\tan x+6 \sin x$.
(ii) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$. [2]
(iii) If $\frac{\mathrm{d} y}{\mathrm{~d} x}=7$ show that $6 \cos ^{3} x-7 \cos ^{2} x+1=0$. [2]
(iv) Hence solve the equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=7$ for $0 \leqslant x \leqslant \pi$ radians. [2]
19 (CIE 2018 , march, paper 12 , question 2)
A curve has equation $y=4+5 \sin 3 x$.
(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$. [2]
(ii) Hence find the equation of the tangent to the curve $y=4+5 \sin 3 x \quad$ at the point where $x=\frac{\pi}{3}$. [3]
20 (CIE 2018, march, paper 12 , question 4 ) It is given that $y=\frac{\ln \left(4 x^{2}-1\right)}{x+2}$.
(i) Find the values of $x$ for which $y$ is not defined.
[2]
(ii) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.
(iii) Hence find the approximate increase in $y$ when $x$ increases from 2 to $2+h$, where $h$ is small. [2]
21 (CIE 2018, s, paper 11, question 11)
The diagram shows part of the graph of $y=16 x+\frac{27}{x^{2}}$, which has a minimum at $A$.
(i) Find the coordinates of $A$. [4]
The points $P$ and $Q$ lie on the curve $\quad y=16 x+\frac{27}{x^{2}}$ and have $x$ -coordinates 1 and 3 respectively.
(ii) Find the area enclosed by the curve and the line $P Q$. You must show all your working. [6]
$22(\mathrm{CIE} 2018, \mathrm{~s}$, paper 12, question 6$)$
Find the coordinates of the stationary point of the curve $y=\frac{x+2}{\sqrt{2 x-1}}$.
$23(\mathrm{CIE} 2018, \mathrm{~s}$, paper 21, question 2
The variables $x$ and $y$ are such that $y=\ln (3 x-1)$ for $x>\frac{1}{3}$.
(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.
(ii) Hence find the approximate change in $x$ when $y$ increases from $\ln (1.2)$ to $\ln (1.2)+0.125$.
24 (CIE 2018, s, paper 21, question 7)
Differentiate with respect to $x$
(i) $4 x \tan x$ [2]
(ii) $\frac{\mathrm{e}^{3 x+1}}{x^{2}-1}$.
25 (CIE 2018, s, paper 22, question 9)
(i) Differentiate $x^{4}(\sqrt{\sin x})$ with respect to $x$.
(ii) Hence find $\int\left(x+\frac{x^{4} \cos x}{\sqrt{\sin x}}+8 x^{3}(\sqrt{\sin x})\right) \mathrm{d} x$.
Answers
1.(i) $\D\frac{3(2-x^2)}{3x+1}-2x\ln(3x+1)$
(ii) $\D \frac{-10x\sec^22x+5\tan2x-20}{25x^2}$
2. $\D x=9$
3. (i) $\D \frac{2x\cos4x+4x^2\sin4x}{\cos^24x}$
(ii) $\D \frac{-\pi}{2}p$
4. (i) $\D -60(3-5x)^{11}$
(ii) $\D x^2\cos x+2x\sin x$
(iii)$\D \frac{(1+e^{2x})\sec^2x-\tan x(2e^{2x})}{(1+e^{2x})^2}$
5. (i)$\D \frac{-1}{x^2}+\frac{1}{\sqrt{x}}$
(ii)$\D \frac{2}{x^3}-\frac{1}{2x^{\frac{3}{2}}}$
(iii) $\D (1,3)$, min
6. (i) $\D \cos\frac{x}{3}\frac{\cos2x}{3} -2\sin(2x)\sin(\frac{x}{3})$
(ii) $\D \frac{(1+\ln x)\sec^2x-\frac{\tan x}{x}}{(1+\ln x)^2}$
7. (i) $\D 4x^3e^{3x}+ 3x^4e^{3x}$
(ii) $\D \frac{-\sin x}{2+\cos x}$
(iii) $\frac{(1+\sqrt{x})\cos x-\sin x(\frac{1}{2\sqrt{x}})}{(1+\sqrt{x})^2}$
8. (a)$\D \frac{(2-x^2)3x^2+2x^4}{(2-x^2)^2}$
(b) $\D k=6$
9. (i) $\D\frac{-5}{2(x-9)^{3/2}}$
(ii) $\D -0.3125h$
10. $\D p=12,q=5$
11. (i) (a) $\D \frac{3e^{3x}}{4x^2+1}- \frac{8xe^{3x}}{(4x^2+1)^2}$
(b) $\D -5,-2$
12. 11/112
13 (i) $\D \frac{2x}{x^2+1}$
(ii) $\D 0.6 h$
14(i) $\D (1+4x)^{10}(-\sin x)+40 (1+4x)^9\cos x$
(ii) $\D \frac{e^{4x-5}(4\tan x)-\sec^2x}{\tan x}$
15. $\D \frac{6}{35}-\frac{4}{25}\ln 14$
16. $y=x^{3}+1 / x-x+2$
17. (i) $\sin x /\left(\cos ^{2} x\right)$
(ii) $\sec ^{2} x+4 \sin x /\left(\cos ^{2} x\right)$
(iii) $\pi / 6,5 \pi / 6$
18. (i) $1,1 / 2,-1 / 3$
(ii) $\sec ^{2} x+6 \cos x$
$(\mathrm{iv}) 0,1.05,1.91$
19. (i) $15 \cos 3 x$
(ii) $y=-15 x+5 \pi+4$
20. (i) $-2,-.5 \leq x \leq .5$
(ii) $\frac{(x+2) 8 x}{4 x^{2}-1}-\ln \left(4 x^{2}-1\right)$
(iii) $4 / 15-\ln (15) / 16$
21. $x=\frac{3}{2}, y=36, \mathrm{~A}=12$
22. $x=3, y=\sqrt{5}$
23. (i) $3 /(3 \mathrm{x}-1)$
(ii) $0.05$
24. (i) $4 \tan x+4 x \sec ^{2} x$
(ii) $\frac{\left(x^{2}-1\right)\left(3 e^{3 x+1}\right)-2 x e^{3 x+1}}{\left(x^{2}+1\right)^{2}}$
25. (i) $4 x^{3} \sqrt{\sin x}$
$+\left(x^{4} / 2\right)\left(\cos x(\sin x)^{-1 / 2}\right) / 2$
(ii) $x^{2} / 2+2 x^{4} \sqrt{\sin x}[+c]$
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