Math topic中文

Product quotient

derivatives of products and quotients of functions

Practice questions

1Given 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…
2A 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…
3Given that $y = \frac{6x^2}{2-5x}$ , find the range of values of $x$ for which $y$ is a decreasing function of $x$ .
4已知函数 $f(x)={{e}^{x}}-1$ ， $g(x)=\sin x(1)$ 判断 $F(x)=f(x)-g(x)$ 在 $x\in[0,+\infty)$ 上零点的个数； (2)当 $x\in[0,\pi]$ 时， $f(x)\geqslant ag(x)(a\in R)$ …
5The 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…
6A curve has the equation $y = \frac{e^{2x+1}}{x+3}$ , where $x \ne -3$ . Find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
7It is given that $y = 4x \sin 2x$ . Show that $\frac{\mathrm{d}y}{\mathrm{d}x}$ can be expressed in the form $ax \cos 2x + b \sin 2x$ …
8Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
9A curve has an equation $y = 2x\sqrt{2x+1}$ . Show that $\frac{dy}{dx} = \frac{ax+b}{\sqrt{2x+1}}$ , where $a$ and $b$ are…
10 $ABCD$ is a rectangle which fits inside a semicircle of radius 8 cm and centre $O$ . It is given that $AB = x$ cm and $BC = y$ …
11Differentiate $4xe^{3x}$ with respect to $x$ . Hence, evaluate $\int_{0}^{3} 6xe^{3x} \text{d}x$ , giving your answer in e…
12A particle moves along the curve $y = \frac{3(x+6)}{x+4}$ , where $x \ne -4$ , in such a way that the $y$ -coordinate of th…
13求下列函数的导数 $(1)y={{x}^{2}}\ln x$ ； $(2)y=\dfrac{\sin x-x}{{{x}^{2}}}$ ； $(3)y={{e}^{-x}}\sqrt{2x-1}$
14The 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 $\pi$ …
15The 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 $y - 3x = 1$ …
16Differentiate $3x \sin 2x$ with respect to $x$ . Hence, evaluate $\int_{0}^{\frac{\pi}{4}} x \cos 2x \text{ d}x$ .
17Given 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 $k$ …
18Answer all the questions A curve has an equation $y = 2x\sqrt{2x+1}$ . Show that $\frac{dy}{dx} = \frac{ax+b}{\sqrt{2x+1}}$ …
19The diagram below shows part of the graph of $y = \frac{x^2 + 2x + 5}{x + 3}$ . Find the coordinates of the minimum point…
20Find the range of values of $x$ such that the graph $y = \ln \left( \frac{x-2}{x-3} \right)^2$ is decreasing.
21Given that $y = x(2x-4)^3$ . find the coordinates of the stationary points of the curve, determine the nature of each of…
22The equation of a curve is $y = \frac{5x}{6 - 2x}$ . Find the $x$ -coordinates of the points at which the gradient of the…
23Show that $\dfrac{d}{dx}(\cot x) = -\csc^{2} x$ . Hence find the value of $\displaystyle \int_{\pi/4}^{\pi/2} -\cot^{2} x\, dx$ …
24Given that $y = \frac{16x}{1+x^2}$ , find the range of values of $x$ for which $y$ is increasing. Another curve has equat…
25Point $P$ lies on the curve $y = \frac{kx^2}{\sqrt{2-x}}$ , where $k$ is a constant. The $x$ -coordinate of $P$ is $-2$ . G…
26Find $\frac{\text{d}}{\text{d}x}(x \cos 2x)$ . Find $\frac{\text{d}}{\text{d}x}(x^2 \sin 2x)$ . Using the results to parts…
27The diagram shows a glass window $ABCDEF$ , consisting of a rectangle $ABEF$ of height $3x\text{ cm}$ and width $y\text{ cm}$ …
28It is given that $y = (x-3)\sqrt{x+1}$ . Find $\dfrac{dy}{dx}$ in the form $\dfrac{ax-b}{2\sqrt{x+1}}$ , where $a$ and $b$ …
29The diagram shows part of the curve $y = \frac{1}{4}x^2 - \frac{1}{2}\ln\left(\frac{x}{3}\right)$ and $P(3,\,2.25)$ is…
30The function is defined by $g(x) = \frac{x^2 - 4}{x^2 + 6}$ , $x > 0$ . Explain, with working, whether $g(x)$ is an increa…
31Given that $y = \frac{3\text{e}^{2x}}{2x+1}$ , find the value of $k$ for which $\frac{\text{d}y}{\text{d}x} = \frac{kxy}{2x+1}$ …
32Find the range of $x$ such that $y = 3x^2 e^{5-2x}$ is a decreasing function.
33Given that $y = e^x (1+\cos x)$ , show $\dfrac{dy}{dx} = e^x (1+\cos x - \sin x)$ . Hence find $\displaystyle \int e^x (\cos x - \sin x)\,dx$ …
34The function $f$ is defined, for all values of $x$ , by $f(x)=(x+3)(1-2x)^2$ . Find the range of values of $x$ for which $f$ …
35Given that $y = \dfrac{4x}{\sqrt{3-2x}}$ , show that $\dfrac{dy}{dx} = \dfrac{12-4x}{\sqrt{(3-2x)^3}}$ . Hence find the va…
36Answer ALL the questions The function $f$ is defined, for all values of $x$ , by $f(x) = (2x - x^2)e^x$ . Find the ran…
37Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(3x^{2}\ln x\right). Hence find$ \int x\ln x,\mathrm{d}x $. The diagram shows the line$ …
38Find $\dfrac{d}{dx}\left(\dfrac{\ln x}{4x}\right). Hence, determine the range of values of$ x $for which the curve$ y=\d…
39Show that $\frac{d}{dx} \left[ (2x)(3-5x)^{\frac{6}{5}} \right] = (6-22x)(3-5x)^{\frac{1}{5}}$ . Hence, evaluate $\int x(3-5x)^{\frac{1}{5}} dx$ …
40设函数 $f(x)=\ln x$ ， $g(x)=\dfrac{x}{x+1}$ ，记 $F(x)=f(x)-g(x)(1)$ 求曲线 $y=f(x)$ 在 $x=e$ 处的切线方程； (2)求函数 $F(x)$ 在 $[\dfrac{1}{e},e^{2}]$ …
41A curve has the equation $y = \frac{e^{2x}}{x-2}$ , where $x > 2$ . Find $\dfrac{dy}{dx}$ . Find the exact value of the coo…
42Given that $y = x^2 \sqrt{2x-1}$ , show that $\frac{\text{d}y}{\text{d}x} = \frac{x(5x-2)}{\sqrt{2x-1}}$ . Determine wheth…
43Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl((2x-3)e^{2x}\bigr). Hence evaluate$ \displaystyle\int_0^1 4xe^{2x},\mathrm{d…
44Find the derivatives of each of the following, simplifying your answer. $y = 3\left(1 - \frac{x}{3}\right)^4$ $f(x) = (2 - 3x)(\sqrt{1 - 4x})$ …
45A particle moves along the curve $y = \frac{2}{\sqrt{3x+1}}$ in such a way that the $y$ -coordinate of the particle is de…
46Differentiate $2x \cos 3x$ with respect to $x$ . Hence, find $\int x \sin 3x \text{ d}x$ .
47The equation of a curve is $y = \frac{x}{e^{2x}}$ . Find an expression for $\frac{dy}{dx}$ . Given that $x$ is changing at…
48Differentiate $x \sin x + \cos x$ with respect to $x$ . Show that $\frac{\text{d}}{\text{d}x} \left( \frac{1}{3} x \sin^3 x \right) = \frac{1}{3} \sin^3 x - x \cos^3 x + x \cos x$ …
49Given that $y = (x + 2)\sqrt{x - 1}$ , show that $\frac{\text{d}y}{\text{d}x} = \frac{kx}{2\sqrt{x - 1}}$ where $k$ is co…
50The equation of a curve is $y = \frac{1 - \sin x}{\cos x}$ . Show that $\frac{dy}{dx} = \tan x \sec x - \sec^2 x$ . Hence…
51[The volume of a pyramid of height $h$ is $\tfrac{1}{3}\times(\text{Base Area})\times h$ .] The diagram shows a piece of…
52The equation of a curve is $y = \frac{\ln x}{x}$ . Find the coordinates of the stationary point of this curve. Leave your…
53The equation of a curve is $y = e^{x^2 - 5x}$ . Find the expressions for $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ . Find th…
54Given that $y = (2x - 1)^3 \ln \sqrt{2x - 1}$ . Show that $\frac{\text{d}y}{\text{d}x} = 6(2x - 1)^2 \ln \sqrt{2x - 1} + (2x - 1)^2$ …
55The function f is defined, for $x > \frac{3}{2}$ , by $\text{f}(x) = \frac{e^{2x}}{1 - x}$ . Explain clearly whether f is…
56Given that $y = \frac{2x^2}{x - 3}$ , determine the range of values of $x$ for which $y$ is a decreasing function.
57It is given that $y = \frac{2x-1}{\sqrt{x+2}}$ for $x > q$ . State the value of $q$ . Find $\dfrac{dy}{dx}$ . State whether…
58A curve has the equation $y = \frac{2 \sin 3x}{2 \cos 3x + 5}$ . Find the value of $a$ and of $b$ for which $\frac{\text{d}y}{\text{d}x} = \frac{a + b \cos 3x}{(2 \cos 3x + 5)^2}$ …
59Show that $\frac{d}{dx}\left(\frac{\ln x}{x^4}\right) = \frac{1}{x^5} - \frac{4\ln x}{x^5}$ . Given that the curve $y = \mathrm{f}(x)$ …
60Given that $y = \frac{\ln(x+3)^3}{3x+9}$ for $x > -3$ , find the set of values of $x$ for which $y$ is an increasing func…