Math topic中文

Chain rule

use of Chain Rule

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…
2In the triangle $ABC$ , $AB = x\ \text{cm}$ , $AC = 3x\ \text{cm}$ and angle $BAC = \dfrac{2\pi}{3}$ . If $x$ is increasing…
3The normal to the curve $y = 2\tan 3x - 1$ at point $P\left(\frac{\pi}{3}, -1\right)$ crosses the $y$ -axis at point $A$ …
4The 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 $Q$ …
5The equation of a curve is given by $y = x + \sin^2 x$ for $0 \le x \le \pi$ . Write down the expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
6The 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…
7A curve has equation $y = \sqrt{ax + (bx)^2} + cx^3 + d$ where $a$ , $b$ , $c$ and $d$ are constants. The curve cuts the x…
8The 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…
9A curve has the equation $y = \frac{e^{2x+1}}{x+3}$ , where $x \ne -3$ . Find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
10The diagram shows part of the curve $y = \sqrt{3x + 4}$ . The line $AB$ is tangent to the curve at $A$ . The points $B$ an…
11It 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 $f''\left(\frac{\pi}{12}\right)$ …
12Given that $y = \frac{4}{5}\left(\frac{x}{12} - 1\right)^6$ and that both $x$ and $y$ vary with time, find the value of…
13When the height of liquid in a container is $h$ m, the volume of liquid is $V\text{ m}^3$ where $V = 0.06 \left[ (4h + 1)^3 - 7 \right] \text{ and } h \ge 0.25$ …
14曲线 $y=\sqrt{3x-2}$ 在点 $(1,f(1))$ 处的切线方程为 A. $x-2y+1=0$ B. $3x-y-2=0$ C. $3x-2y-1=0$ D. $3x+2y-5=0$
15Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
16It 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…
17A particle moves in a straight line, so that, $t$ seconds after leaving a fixed point $O$ its velocity, $v$ is given by…
18Differentiate $\ln\left(\frac{3x}{x^2+1}\right)$ with respect to $x$ . Hence find $\displaystyle \int \frac{2x}{x^2+1}\,dx$ …
19The diagram below shows the graph of $y = \frac{1}{2x+1}$ and the tangent to the curve where the curve intersects the $y$ …
20A 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…
21Show that $\frac{d}{dx}(\tan^{3} 2x) = 6\sec^{4} 2x - 6\sec^{2} 2x$ . Hence, find the exact value of $\displaystyle \int_{0}^{\pi/8} \sec^{4} 2x\, dx$ …
22 $ABCD$ is a rectangle which fits inside a semicircle of radius 8 cm and centre $O$ . It is given that $AB = x$ cm and $BC = y$ …
23The diagram shows part of the curve $y = \frac{1}{\left(\frac{3}{4}x - 5\right)^2} + 1$ . $P$ is the point on the curve w…
24求下列函数的导数 $(1)y={{x}^{2}}\ln x$ ； $(2)y=\dfrac{\sin x-x}{{{x}^{2}}}$ ； $(3)y={{e}^{-x}}\sqrt{2x-1}$
25Given 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 $k$ …
26Answer all the questions A curve has an equation $y = 2x\sqrt{2x+1}$ . Show that $\frac{dy}{dx} = \frac{ax+b}{\sqrt{2x+1}}$ …
27A curve is defined by the equation $y = \ln\left(\frac{\text{e}^x}{x^2 + 10}\right)$ . Show that the curve has no turning…
28The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
29Find the range of values of $x$ such that the graph $y = \ln \left( \frac{x-2}{x-3} \right)^2$ is decreasing.
30[The volume of a cone of height $h$ and base radius $r$ is $\frac{1}{3}\pi r^2 h$ .] A hollow right circular cone of base…
31Given that $y = x(2x-4)^3$ . find the coordinates of the stationary points of the curve, determine the nature of each of…
32The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
33Find the derivative of the functions $y = \ln x$ and $y = \ln 2x$ respectively. Explain why, in (i), having the same der…
34求 $f(x) = \ln(x^2 + 1)$ 的导数 $f'(x)$ 。
35The graph shows part of the curve $y = -3(2x-4)^{-1/3}$ . The point $P(-2,\ 1.5)$ lies on the curve and the normal to the…
36The diagram shows part of the curve $y = \sqrt{2x + 4}$ , meeting the $x$ -axis at the point $A$ and the $y$ -axis at the p…
37The diagram shows part of the curve $y = 2 \sin 2x + 1$ . $P$ is the maximum point of the curve and $Q$ is the point on t…
38A curve has the equation $y = 3(4 - x)^{3} + 5$ . Find the coordinates of the stationary point of the curve. Determine th…
39Point $P$ lies on the curve $y = \frac{kx^2}{\sqrt{2-x}}$ , where $k$ is a constant. The $x$ -coordinate of $P$ is $-2$ . G…
40Find $\frac{\text{d}}{\text{d}x}(x \cos 2x)$ . Find $\frac{\text{d}}{\text{d}x}(x^2 \sin 2x)$ . Using the results to parts…
41The diagram shows a glass window $ABCDEF$ , consisting of a rectangle $ABEF$ of height $3x\text{ cm}$ and width $y\text{ cm}$ …
42It 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 $b$ …
43The 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…
44The diagram shows part of the curve $y = 27 + \frac{8}{(x-2)^3}$ which crosses the axes at points $A$ and $B$ . Explain w…
45Find the range of $x$ such that $y = 3x^2 e^{5-2x}$ is a decreasing function.
46The 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 $f$ …
47Given that $y = \dfrac{4x}{\sqrt{3-2x}}$ , show that $\dfrac{dy}{dx} = \dfrac{12-4x}{\sqrt{(3-2x)^3}}$ . Hence find the va…
48Express $\dfrac{8 - 3x}{(2x + 3)(x^2 + 4)}$ in partial fractions. Differentiate $\ln\left(x^2 + 4\right)$ with respect t…
49Show that $\frac{d}{dx} \left[ (2x)(3-5x)^{\frac{6}{5}} \right] = (6-22x)(3-5x)^{\frac{1}{5}}$ . Hence, evaluate $\int x(3-5x)^{\frac{1}{5}} dx$ …
50The diagram shows part of the curve $y = \frac{7}{6x + 1}$ intersecting the $y$ -axis at $A(0, 7)$ . The tangent to the cu…
51A particle moves in a straight line such that, $t$ s after passing through a fixed point $O$ , its displacement from $O$ …
52A particle $P$ travels in a straight line and its velocity, $v$ m/s, is given by $v = 4\sin^2\left(\frac{t}{2}\right) - 1$ …
53Given that $y = \sec x$ , show that $\frac{dy}{dx} = \sec x \tan x$ . Hence show that $y^4 - y^2 = \left(\frac{dy}{dx}\right)^2$ …
54Given 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…
55Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl((2x-3)e^{2x}\bigr). Hence evaluate$ \displaystyle\int_0^1 4xe^{2x},\mathrm{d…
56A particle moves from point P towards point O. The speed of the particle, $t$ seconds after passing point P, can be mode…
57Find the derivatives of each of the following, simplifying your answer. $y = 3\left(1 - \frac{x}{3}\right)^4$ $f(x) = (2 - 3x)(\sqrt{1 - 4x})$ …
58A particle moves along the curve $y = \frac{2}{\sqrt{3x+1}}$ in such a way that the $y$ -coordinate of the particle is de…
59Differentiate $2x \cos 3x$ with respect to $x$ . Hence, find $\int x \sin 3x \text{ d}x$ .
60Differentiate $x \sin x + \cos x$ with respect to $x$ . Show that $\frac{\text{d}}{\text{d}x} \left( \frac{1}{3} x \sin^3 x \right) = \frac{1}{3} \sin^3 x - x \cos^3 x + x \cos x$ …