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

Expansions

For questions 1-9 suppose that $$f\left(x\right)=c_0+c_1\left(x-a\right)+c_2{\left(x-a\right)}^2+c_3{\left(x-a\right)}^3+c_4{\left(x-a\right)}^4+\dots$$

	Show that $c_0=f\left(a\right).$
	Show that ${\displaystyle c_1=\frac{df}{dx}{|}_{x=a}}.$
	Show that ${\displaystyle c_2=\frac{1}{2}\left(\frac{d^{2}f}{d{x}^2}{|}_{x=a}\right)}.$
	Show that ${\displaystyle c_3=\frac{1}{3!}\left(\frac{d^{3}f}{d{x}^3}{|}_{x=a}\right)}.$
	Show that ${\displaystyle c_4=\frac{1}{4!}\left(\frac{d^{4}f}{d{x}^4}{|}_{x=a}\right)}.$
	Show that ${\displaystyle c_5=\frac{1}{5!}\left(\frac{d^{5}f}{d{x}^5}{|}_{x=a}\right)}.$
	Explain why ${\displaystyle c_n=\frac{1}{n!}\left(\frac{d^{n}f}{d{x}^n}{|}_{x=a}\right)}.$
	Show that $$f\left(a+\Delta x\right)=f\left(a\right)+\left(\frac{df}{dx}{|}_{x=a}\right)\left(\Delta x\right)+\frac{1}{2}\left(\frac{d^{2}f}{d{x}^2}{|}_{x=a}\right){\left(\Delta x\right)}^2+\frac{1}{3!}\left(\frac{d^{3}f}{d{x}^3}{|}_{x=a}\right){\left(\Delta x\right)}^3+\frac{1}{4!}\left(\frac{d^{4}f}{d{x}^4}{|}_{x=a}\right){\left(\Delta x\right)}^4+\dots$$
	Show that ${\displaystyle \underset{\Delta x\to 0}{\lim }\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x}=\frac{df}{dx}{|}_{x=a}}.$
	Give a series expansion for ${\displaystyle e^x}$
	Give a series expansion for ${\displaystyle \sin x}$
	Give a series expansion for ${\displaystyle \cos x}$
	Give a series expansion for ${\displaystyle \frac{1}{1-x}}$
	Give a series expansion for ${\displaystyle \frac{1}{1+x}}$
	Give a series expansion for ${\displaystyle \frac{1}{1+x^2}}$
	Explain why ${\displaystyle 1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\dots =\frac{3}{2}.}$
	Explain why ${\displaystyle 1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\dots +\frac{1}{3^{50}}=\frac{3}{2}-\frac{1}{2\times 3^{50}}.}$

Derivatives at a point

	Let $y=\tan 2x-2\tan x+2.$ Find ${\displaystyle \frac{dy}{dx}}$ at $x=\pi /4.$
	Let $y=\frac{{\sin }^{2}x+\cos x}{1+x^2}.$ Find ${\displaystyle \frac{dy}{dx}{|}_{x=0}}$ and ${\displaystyle \frac{dy}{dx}{|}_{x=\frac{\pi }{2}}.}$
	Let $y=\cos \left(\sin x^2\right).$ Find ${\displaystyle \frac{dy}{dx}{|}_{x=\frac{\pi }{3}}.}$
	Let ${\displaystyle y={\left(\cot \sqrt{x}+5{\sin }^2\sqrt{x}\right)}^2.}$ Find ${\displaystyle \frac{dy}{dx}}$ at $x={\pi }^2/16.$
	Let ${\displaystyle y=\frac{\sin x^2}{\sqrt{1+x^2}}.}$ Find ${\displaystyle \frac{dy}{dx}{|}_{x=0}}$ and ${\displaystyle \frac{dy}{dx}{|}_{x=\sqrt{\frac{\pi }{2}}}.}$

Differential equations

	If $y=\tan x+x$ show that ${\displaystyle {\cos }^{2}x\times \frac{d^{2}y}{d{x}^2}-2y+2x=0}.$
	If $y=A\cos nx+B\sin nx$ show that ${\displaystyle >\frac{d^{2}y}{d{x}^2}+n^{2}y=0}.$
	If $y=2\sin x+3\cos x$ show that ${\displaystyle y+\frac{d^{2}y}{d{x}^2}=0}.$
	If $y=a\sin x+b\cos x$ show that ${\displaystyle y+\frac{d^{2}y}{d{x}^2}=0}.$
	If $y=\sin \left(\sin x\right)$ show that ${\displaystyle y+\frac{d^{2}y}{d{x}^2}+\tan x\frac{dy}{dx}+y{\cos }^{2}x=0}.$
	If $y=a\sin x+b\cos x$ show that ${\displaystyle y^2+{\left(\frac{d^{2}y}{d{x}^2}\right)}^2=a^2+b^2}.$
	If $y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\sqrt{\dots }}}}$ show that ${\displaystyle \left(2y-1\right)\frac{dy}{dx}=\cos x}.$

Parametric equations.

	Find ${\displaystyle \frac{dy}{dx}}$ when $x=a\cos \theta$ and $y=b\sin \theta .$
	Find ${\displaystyle \frac{dy}{dx}}$ when $x=a\left(\theta +\sin \theta \right)$ and $y=a\left(1-\cos \theta \right).$
	Find ${\displaystyle \frac{dy}{dx}}$ when $x=a{\sec }^2\theta$ and $y=b{\tan }^3\theta .$
	Find ${\displaystyle \frac{dy}{dx}}$ when $x=b{\sin }^3\theta$ and $y=b{\cos }^3\theta .$
	If $x=a\left(t-\sin t\right)$ and $y=a\left(1-\cos t\right)$ find ${\displaystyle \frac{d^{2}y}{d{x}^2}}.$
	If $x=a\left(\theta +\sin \theta \right)$ and $y=a\left(1-\cos \theta \right)$ find ${\displaystyle \frac{d^{2}y}{d{x}^2}}$ at $\theta =\pi /2.$
	If $x=2\cos \theta -\cos 2\theta$ and $y=2\sin \theta -\sin 2\theta$ find ${\displaystyle \frac{d^{2}y}{d{x}^2}}.$
	If $x=\sin t$ and $y=\sin mt$ prove that ${\displaystyle \left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+m^{2}y=0}.$

Implicit differentiation.

	Find ${\displaystyle \frac{dy}{dx}}$ when $y^2\sin x+y\tan x+\left(1+x^2\right)\cos x=0.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $\sin \left(xy\right)+x/y=x^2-y.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle \frac{y}{1+x^2}+{\tan }^{2}x+\sin y=0.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle \tan \left(x+y\right)+\tan \left(x-y\right)=1.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle a\sin \left(xy\right)+b\cos \left(x,/,y\right)=0.}$
	If $x=\ln \left(\tan \left(y/x\right)\right)$ find ${\displaystyle \frac{dy}{dx}}.$

Derivatives with inverse trig functions.

	Find ${\displaystyle \frac{dy}{dx}}$ when $y={\sin }^{-1}{x}^3.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y={\tan }^{-1}\sqrt{x}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y={\sin }^{-1}3x.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y={\csc }^{-1}{x}^2.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y={\cos }^{-1}\sqrt{x}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y={\csc }^{-1}\left(\sin x\right).$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y={\tan }^{-1}\sqrt{x-1}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y=\sin \left({\tan }^{-1}x\right).$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y=x{\cos }^{-1}x.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y=x{\sin }^{-1}x.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y={\tan }^{-1}\sqrt{x}-{\tan }^{-1}x.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y=\left(1+x^2\right){\tan }^{-1}x.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y=\tan x{\cos }^{-1}x.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)+{\mathrm{tsn}}^{-1}x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle \left(1-x^2\right){\cos }^{-1}x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\tan x\times {\tan }^{-1}x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sec }^{-1}x+{\csc }^{-1}x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^{-1}\left(\frac{a}{x}\right)\times {\cot }^{-1}\left(\frac{x}{a}\right).}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left({\tan }^{-1}2x\right)}^3.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\cos }^{-1}\left(\tan x^2\right).}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^{-1}\left(\frac{2x}{1-x^2}\right).}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sec }^{-1}\left(\frac{1-x^2}{1+x^2}\right).}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right).}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\tan }^{-1}\left(\frac{1+x^2}{1-x^2}\right).}$
	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 ${\displaystyle y={\cot }^{-1}{\left(\frac{1+\cos x}{1-\cos x}\right)}^{\frac{1}{2}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\cot }^{-1}{\left(\frac{1+\cos 3x}{1-\cos 3x}\right)}^{\frac{1}{2}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\sin }^{-1}{\left(\frac{a+b\cos x}{b+a\cos x}\right)}^{\frac{1}{2}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\cot }^{-1}{\left(\frac{1+2\cos x}{2+\cos x}\right)}^{\frac{1}{2}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\cot }^{-1}{\left(\frac{1-\cos x}{\sin x}\right)}^{\frac{1}{2}}.}$
	Differentiate ${\displaystyle {\sin }^{-1}\left(\frac{x^2-1}{1+x^2}\right)}$ with respect to ${\displaystyle {\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)}$
	If ${\displaystyle y=\frac{{\sin }^{-1}x}{\sqrt{1-x^2}}}$ prove that ${\displaystyle \left(1-x^2\right)\frac{dy}{dx}-xy=1.}$

Derivatives with trigonometric functions

	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x\cos x-\sin x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\cos }^{3}3x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(x^2+\cos x\right)}^4.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sin x\sin 2x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{\sin 2x}{x^2}.}$

Derivatives with exponentials and logs

	Find ${\displaystyle \frac{dy}{dx}}$ when $y=\ln \left(x+\sqrt{x^2+a^2}\right).$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{1+e^{\mathrm{x}}}{1-e^x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \left(\frac{x^2+x+1}{x^2-x-1}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \left[e^x{\left(\frac{x-2}{x+2}\right)}^{\frac{3}{4}}\right]}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \ln \ln x^4}.$

Derivatives with exponentials, logs and trig functions.

	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=a^{\cos 1}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \frac{{\sin }^{m}x}{{\cos }^{n}x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=e^{ax}\sin bx}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \left(\frac{1-\cos x}{1+\cos x}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \sqrt{\frac{1-\tan x}{1+\tan x}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=e^{ax}\cos \left(bx+c\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{\sqrt{x+\ln \tan x}}{x{e}^{2x}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \frac{1+x\sin x}{1-x\sin x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln {\left(\frac{1-\cos x}{1+\cos x}\right)}^{\frac{1}{2}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \sqrt{\frac{1+\sin x}{1-\sin x}}}.$
	If $y=\ln \left(\sin x\right)$ show that ${\displaystyle \frac{d^{3}y}{d{x}^3}=2{\csc }^{2}x\cot x}.$
	If $y=e^{ax}\cos bx$ show that ${\displaystyle \frac{d^{2}y}{d{x}^2}-2a\frac{dy}{dx}+\left(a^2+b^2\right)y=0.}$
	If $y=a\cos \left(\ln x\right)+b\sin \left(\ln x\right)$ show that ${\displaystyle x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0.}$
	If $y=A{e}^{-kt}\cos \left(pt+c\right)$ show that ${\displaystyle \frac{d^{2}y}{d{t}^2}+2k\frac{dy}{dt}+n^{2}y=0,}$ where $n^2=p^2+k^2$
	If $y=e^{-x}\cos x$ show that ${\displaystyle \frac{d^{4}y}{d{x}^4}+4y=0.}$

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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