Problem Set - Differentiation

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 1 March 2010

Derivatives with exponentials, logs and trig functions.

	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{2\tan x}{\tan x+\cos x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sqrt{x\sin x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{x+\sin 2x}{\cos 3x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=e^{5x}\ln \left(\sec x\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{x^5}{{\sin }^{-1}2x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sin x^2-\frac{\tan x}{1+x^2}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\tan \sqrt{x}+x^2-\sin x\right)}^3}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{{\sin }^{3}x{\cos }^{3}x}{\cos 3x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=e^x\tan x+\frac{\ln x}{\sin x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=2a^x\ln x}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{x^{\frac{3}{2}}+1}{3\sqrt{x}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{1-x^2}{x^2-x+1}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sqrt{1+\ln x\sin x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=7x^{\frac{1}{2}}+5x^{-\frac{7}{2}}+{\sin }^{-1}{x}^4-\ln \cot x}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sin }^{2}x{\cos }^{3}x}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sin mx\cos nx}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sin }^{m}x{\cos }^{n}x}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\cos }^{-1}\left(1-2x^2\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sin }^{-1}\left(3x-4x^3\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{\sqrt{a+x}-\sqrt{a-x}}{\sqrt{a+x}+\sqrt{a-x}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\left(1+x\right)\left(1+2x\right)\left(1+3x\right)\left(1+4x\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^2\sqrt{1-x^2}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{\tan x}{x^2}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{e^{2x}}{\ln x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{e^{x^2}{\tan }^{-1}x}{\sqrt{1+x^2}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=e^{\sqrt{x}+2}-e^{\sqrt{x+2}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=7^{x^2+2x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={cot}^2\left(e^{3x}{x}^x\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \left({\tan }^{-1}x\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\csc }^{-1}\left(\frac{1+x^2}{2x}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sin }^{-1}x+{\sin }^{-1}\sqrt{1-x^2}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sec }^{-1}\left(\frac{x^2+1}{x^2-1}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x{\sin }^{-1}x+\sqrt{1-x^2}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x{\cos }^{-1}2x-\frac{1}{2}\sqrt{1-4x^2}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{1}{2}{\tan }^{-1}\left(\frac{1}{2}\tan \left(\frac{x}{2}\right)\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^{-1}\left(\sec x+\tan x\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{x{\cos }^{-1}x}{\sqrt{1-x^2}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{1}{2}x\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}\frac{x}{a}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sin }^{-1}\left(2x\sqrt{1-x^2}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x^3\sin 2x+\frac{\cos x}{x+1}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x^4\sin 2x+\frac{x^2}{x^3+1}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x^{\ln x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\tan x\right)}^{\cot x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\tan x\right)}^{\cot x}+{\left(\cot x\right)}^{\tan x}+x^{{\cos }^{-1}x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\left(\sin x\right)\left(e^x\right)\left(\ln x\right)\left(x^x\right)\left(x^{{\cos }^{-1}},x\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x^y=y^x}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle e^{xy}-4xy=0}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle xy=\sin \left(x+y\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle \frac{x^m}{a^m}+\frac{y^m}{b^m}=1}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x^m{y}^n={\left(x+y\right)}^{m+n}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y\ln x=x-y}.$

Integrals with mixed functions.

	${\displaystyle \int \frac{{\sin }^{-1}x}{\sqrt{1-x^2}}dx}$
	${\displaystyle \int \frac{\sin \left(2{\tan }^{-1}x\right)}{1+x^2}dx}$
	${\displaystyle \int \frac{\cos \left(\ln x\right)}{x}dx}$
	${\displaystyle \int \frac{{\csc }^2\left(\ln x\right)}{x}dx}$
	${\displaystyle \int e^{\tan x}{\sec }^{2}xdx}$
	${\displaystyle \int e^{{\cos }^{2}x}\sin 2xdx}$
	${\displaystyle \int \cot x\ln \left(\sin x\right)dx}$
	${\displaystyle \int \frac{\cot x}{\ln \left(\sin x\right)}dx}$
	${\displaystyle \int \sec x\ln \left(\sec x+\tan x\right)dx}$
	${\displaystyle \int \frac{x{\tan }^{-1}{x}^2}{1+x^4}dx}$
	${\displaystyle \int \frac{x{\sin }^{-1}{x}^2}{\sqrt{1-x^4}}dx}$
	${\displaystyle \int \frac{1}{\sqrt{1-x^2{\sin }^{-1}x}}dx}$
	${\displaystyle \int \frac{1+\tan x}{x+\ln \left(\sec x\right)}dx}$
	${\displaystyle \int \frac{\sec x\csc x}{\ln \left(\tan x\right)}dx}$
	${\displaystyle \int \frac{1}{x{\cos }^2\left(1+\ln x\right)}dx}$
	${\displaystyle \int e^{-x}{\csc }^2\left(2e^{-x}+5\right)dx}$
	${\displaystyle \int x^2{e}^{x^3}\cos e^{x^3}dx}$
	${\displaystyle \int \frac{e^{m{\tan }^{-1}x}}{1+x^2}dx}$
	${\displaystyle \int \frac{\left(x+1\right)e^x}{{\cos }^2\left(x{e}^x\right)}dx}$
	${\displaystyle \int \frac{e^{\sqrt{x}}\cos e^{\sqrt{x}}}{\sqrt{x}}dx}$

Areas of regions.

	Find the area inside the ellipse ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1.}$
	Using integration find the area of the triangle with vertices (-1,1), (0,5) and (3,2).
	Graph the region $\left\{\left(x,y\right)|4x^2+9y^2\le 36\right\}$ and find its area.
	Find the area of the region $\left\{\left(x,y\right)|y^2\le 8x,x^2+y^2\le 9\right\}.$
	Find the area of the region $\left\{\left(x,y\right)|y^2\le x,x^2+y^2\le 2\right\}.$
	Find the area of the region $\left\{\left(x,y\right)|y^2\ge ax,x^2+y^2\le 2ax,x\ge 0,y\ge 0\right\}.$
	Find the area of the region $\left\{\left(x,y\right)|y^2\le 4x,4x^2+4y^2\le 9\right\}.$
	Find the area of the region $\left\{\left(x,y\right)|x^2+y^2\le 1\le x+y\right\}.$
	Find the area of the region $\left\{\left(x,y\right)|0\le y\le x^2+1,0\le y\le x+1,0\le x\le 2\right\}.$
	Find the area of the region $\left\{\left(x,y\right)|x^2\le y\le \left|x\right|\right\}.$

Different types of volume problems.

	A solid is generated by rotating, sbout the $x$-axis, the area bounded by the curve $y=f\left(x\right)$, the $x$-axis, and the lines $x=a,x=b.$ Its volume, for all $b>a,$ is $b^2-ab.$ Find $f\left(x\right).$
	A solid is generated by rotating the curve $y=f\left(x\right),0\le x\le a,$ about the $x$-axis. Its volume, for all $a,$ is $a+a^2.$ Find $f\left(x\right).$
	The area bounded by the curve $y^2=4x$ and the straight line $y=x$ is rotated about the $x$-axis. Find the volume generated.
	Sketch the area bounded by the curve $y^2=4ax,$ the line $x=a,$ and the $x$-axis. Find the volumes generated by rotating this area in each of the following ways: about the $x$-axis. about the line $x=a$ about the $y$-axis.
	The area bounded by the curve $y=\frac{x}{\sqrt{x^3+8}},$ the $x$-axis, and the line $x=2$ i rotated about the $y$-axis. Compute the volume.
	Find the volume of the solid generated by rotating the larger area bounded by $y^2=x-1,x=3,$ and $y=1$ about the $y$-axis.
	The area bounded by the curve $y^2=4ax$ and the line $x=a$ is rotated about the line $x=2a.$ Find the volume generated.
	A twisted solid is generated as follows: We are given a fixed line $L$ in space, and a square of side length $s$ in a plane perpendicular to $L.$ One vertex of the square is on $L.$ As this vertex moves a distance $h$ on $L,$ the square turns through one full revolution, with $L$ as the axis. Find the volume generated.
	A twisted solid is generated as follows: We are given a fixed line $L$ in space, and a square of side length $s$ in a plane perpendicular to $L.$ One vertex of the square is on $L.$ As this vertex moves a distance $h$ on $L,$ the square turns through two full revolutions, with $L$ as the axis. Find the volume generated.
	Two circles have a common diameter and lie in perpendicular planes. A square moves so that its plane is perpendicular to this diameter and its diagonals are chords of the circles. Find the volume generated.
	Find the volume generated by rotating the area bounded by the $x$-axis and one arch of the curve $y=\sin 2x$ about the $x$-axis.
	A round hole of radius $\sqrt{3}$ ft is bored through the center of a solid sphere of radius 2 ft. How much is the volume cut out?

References

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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