Math topic中文

Standard derivatives

derivatives of $x^{n}$ for any rational $n$ , $\sin x$ , $\cos x$ , $\tan x$ , $e^{x}$ , and $\ln x$ together with constant multiples, sums and differences

Practice questions

1The diagram shows a greenhouse built on a rectangular base. $140\text{ m}^2$ of polyethene is used to build the curved r…
2Given that $y = \dfrac{3\ln 2x}{x^2}$ for $x>0$ . Show that $\dfrac{dy}{dx} = \dfrac{3}{x^3}\left(1 - 2\ln 2x\right).$ He…
3In the triangle $ABC$ , $AB = x\ \text{cm}$ , $AC = 3x\ \text{cm}$ and angle $BAC = \dfrac{2\pi}{3}$ . If $x$ is increasing…
4A curve has the equation $y = \frac{2x - 3}{3x + 4}$ , where $x \ne -\frac{4}{3}$ . The normal to the point $P$ on the cur…
5Find the gradient function of the curve $y = 2x^2 - \ln x + c$ . The curve touches the positive $x$ -axis at point $T$ . Fi…
6The normal to the curve $y = 2\tan 3x - 1$ at point $P\left(\frac{\pi}{3}, -1\right)$ crosses the $y$ -axis at point $A$ …
7The point $P$ lies on the curve $y = \ln\left(\frac{x+1}{x-1}\right)$ for $x > 1$ . The normal to the curve at $P$ is par…
8The radius of a circle increases at a rate of 3 cm s^{-1}. Find, in terms of $\pi$ , the rate of increase of the area whe…
9The figure shows part of the curve $y = g(x)$ . $(2, 8)$ and $(6, 1)$ are two points on the curve. Given that $\int_{2}^{6} y \text{ } dx = 32$ …
10The diagram shows part of the curve $y = 3e^{-\frac{1}{2}x} - 6$ which crosses the $x$ -axis at $P$ and the $y$ -axis at $Q$ …
11The equation of a curve is given by $y = x + \sin^2 x$ for $0 \le x \le \pi$ . Write down the expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
12The volume of liquid in a container, $V\ \text{m}^3$ , is given by $V = 0.05\bigl[(3x+2)^3 - 8\bigr],$ where $x$ is the…
13A motorist, travelling at a constant velocity of V m/s, passed a fixed point X and saw few vehicles ahead. He stepped on…
14The equation of a curve is given by $y = 5xe^{2x+1}$ . Find $\frac{dy}{dx}$ . Hence, find $\int xe^{2x+1} \text{ d}x$ . Det…
15A curve has the equation $y = \frac{e^{2x+1}}{x+3}$ , where $x \ne -3$ . Find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
16A cone of height, $h$ and radius, $r$ fits exactly inside a sphere of radius $5\text{ cm}$ as shown below. $\left(\text{Volume of cone} = \frac{1}{3}\pi r^2 h\right)$ …
17The diagram shows part of the curve $y = \sqrt{3x + 4}$ . The line $AB$ is tangent to the curve at $A$ . The points $B$ an…
18It is given that $f'(x)=6\cos 3x-12\sin 3x$ and $f\left(\frac{\pi}{3}\right)=0$ . Find $f(x)$ . Find the exact value of $f''\left(\frac{\pi}{12}\right)$ …
19A particle, P, starts moving in a straight line with a velocity of 8 m/s at a displacement of 3 m from a fixed point O.…
20It is given that $y = 4x \sin 2x$ . Show that $\frac{\mathrm{d}y}{\mathrm{d}x}$ can be expressed in the form $ax \cos 2x + b \sin 2x$ …
21Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
22A conical vessel, whose vertical angle is $30^\circ$ as shown in the diagram, has water poured into it at such a rate th…
23The diagram shows part of the curve $y = (1 - x)(x - 4)$ and two lines $l_1$ and $l_2$ . The lines $l_1$ and $l_2$ are ta…
24It is given that $f(x)$ is such that $f'(x) = \sin 2x + \cos 3x$ . Given also that $f\left(\frac{\pi}{6}\right) = 0$ , sho…
25A particle $P$ moves in a straight line so that, $t$ seconds after passing through a fixed point $O$ , its velocity $v$ m…
26Given the curve $y = \frac{1}{2}e^{2x} - \frac{1}{3}e^{-3x} + \frac{1}{3}$ and the normal to the curve at point $P\left(\ln 2, \frac{55}{24}\right)$ …
27Differentiate $\ln\left(\frac{3x}{x^2+1}\right)$ with respect to $x$ . Hence find $\displaystyle \int \frac{2x}{x^2+1}\,dx$ …
28The diagram shows the glass test tube of radius $r\text{ cm}$ and height $(h+r)\text{ cm}$ with a hemisphere of radius $r\text{ cm}$ …
29Show that $\frac{d}{dx}(\tan^{3} 2x) = 6\sec^{4} 2x - 6\sec^{2} 2x$ . Hence, find the exact value of $\displaystyle \int_{0}^{\pi/8} \sec^{4} 2x\, dx$ …
30A farmer uses 160 m of fencing to enclose a plot of his land in a shape that comprises an isosceles triangle and a recta…
31Differentiate $4xe^{3x}$ with respect to $x$ . Hence, evaluate $\int_{0}^{3} 6xe^{3x} \text{d}x$ , giving your answer in e…
32Oil is dripping into an empty inverted right circular cone at the rate of 16 cm3 per second. The height, h cm, of the co…
33A factory manufactures miniature containers that are made up of a closed circular cylinder and a hemisphere which have t…
34求下列函数的导数 $(1)y={{x}^{2}}\ln x$ ； $(2)y=\dfrac{\sin x-x}{{{x}^{2}}}$ ； $(3)y={{e}^{-x}}\sqrt{2x-1}$
35The equation of a curve is $y = e^{-2x}\tan x$ . Show that $\dfrac{dy}{dx} = e^{-2x}(1 - \tan x)^2$ . Find, in terms of $\pi$ …
36The point A lies on the curve $y = x \ln x^2$ , $x > 0$ . The tangent to the curve at A is parallel to the line $y - 3x = 1$ …
37Differentiate $3x \sin 2x$ with respect to $x$ . Hence, evaluate $\int_{0}^{\frac{\pi}{4}} x \cos 2x \text{ d}x$ .
38Given that $y = 2\cos x \sin 2x - \sin x \cos 2x$ , show that $\dfrac{dy}{dx} = k\cos x \cos 2x$ , stating the value of $k$ …
39A curve has the equation $y = x^3 + kx^2 + kx + 8$ . Find the set of values of $k$ such that the curve is an increasing f…
40A particle passes a fixed point $O$ and moves in a straight line such that, $t$ s after leaving $O$ , its velocity, $v\text{ m/s}$ …
41[The volume of a sphere of radius r is $\dfrac{4}{3}\pi r^3$ , and its surface area is $4\pi r^2$ .] The diagram shows a b…
42Given that $y = 3e^{2x} + 2e^{-x}$ , and that $\frac{\text{d}^2 y}{\text{d}x^2} + \frac{\text{d}y}{\text{d}x} + y = Ae^{2x} + Be^{-x}$ …
43A curve is defined by the equation $y = \ln\left(\frac{\text{e}^x}{x^2 + 10}\right)$ . Show that the curve has no turning…
44Air is being pumped into a spherical balloon at a constant rate of 12 cm $^3$ per second. Find the rate at which the radi…
45The diagram shows part of the curve $y = \sin x$ . A is a point on the curve that lies on the line $x = \frac{\pi}{3}$ . T…
46The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
47The diagram below shows part of the graph of $y = \frac{x^2 + 2x + 5}{x + 3}$ . Find the coordinates of the minimum point…
48A cone of height 12 cm and base radius 5 cm is placed over a cylinder of radius $r$ cm and height $h$ cm. The cone is in…
49The diagram shows part of the curve $y = e^{x-1} - 2$ . The point $P(2,\ e-2)$ lies on the curve and the normal to the cu…
50Find the range of values of $x$ such that the graph $y = \ln \left( \frac{x-2}{x-3} \right)^2$ is decreasing.
51The diagram shows a prism in which the cross-section is a trapezium $ABCD$ . $AB = 4x\text{ cm}$ , $BC = DC = AD = 2x\text{ cm}$ …
52画出曲线 $y = x^3 - 4x$ 的草图，标出零点、极值点和拐点坐标。
53A miniature model in the shape of a triangular prism is shown above. A piece of wire, 36 cm in length, runs through all…
54A particle P moves along a horizontal straight line so that t seconds after motion has begun, its displacement, s m, fro…
55A particle moves in a straight line so that its velocity, $v$ m/s is given by $v = t^2 - 4t + 3$ , where $t$ is the time…
56A particle travels in a straight line so that, $t$ seconds after leaving a fixed point $O$ , its velocity, $v\text{ m s}^{-1}$ …
57Find the derivative of the functions $y = \ln x$ and $y = \ln 2x$ respectively. Explain why, in (i), having the same der…
58It is given that $y = x^3 + px^2 + qx + 10$ where $p$ and $q$ are integers. The only values of $x$ for which $y$ is a de…
59求 $f(x) = \ln(x^2 + 1)$ 的导数 $f'(x)$ 。
60Baking powder is poured onto a flat surface at a constant rate of $2\pi\text{ cm}^3\text{s}^{-1}$ , forming a right circu…