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

The chain rule

	Let $g$ be a function. Show that ${\displaystyle \frac{d{g}^0}{dx}=0\frac{dg}{dx}}$.
	Let $g$ be a function. Show that ${\displaystyle \frac{d{g}^1}{dx}=1g^0\frac{dg}{dx}}$.
	Let $g$ be a function. Show that ${\displaystyle \frac{d{g}^2}{dx}=2g^1\frac{dg}{dx}}$.
	Let $g$ be a function. Show that ${\displaystyle \frac{d{g}^3}{dx}=3g^2\frac{dg}{dx}}$.
	Let $g$ be a function. Show that ${\displaystyle \frac{d{g}^4}{dx}=4g^3\frac{dg}{dx}}$.
	Let $g$ be a function. Show that ${\displaystyle \frac{d{g}^5}{dx}=5g^4\frac{dg}{dx}}$.
	Let $g$ be a function. Show that ${\displaystyle \frac{d{g}^n}{dx}=n{g}^{n-1}\frac{dg}{dx}}$.
	Let $f\left(y\right)=4y^3+7y^2+2y-13$ and let $g$ be a function. Show that ${\displaystyle \frac{d\left(f\left(g\right)\right)}{dx}=\left(12g^2+14g+2\right)\frac{dg}{dx}\text{.}}$
	Let $f$ be a polynomial and let $g$ be a function. Show that ${\displaystyle \frac{d\left(f\left(g\right)\right)}{dx}=\frac{df}{dg}·\frac{dg}{dx}}$.

Derivatives of the basic functions

	Explain why ${\displaystyle \frac{d{e}^x}{dx}=e^x}$.
	Explain why ${\displaystyle \frac{d\sin x}{dx}=\cos x}$.
	Explain why ${\displaystyle \frac{d\cos x}{dx}=-\sin x}$.
	Explain why ${\displaystyle \frac{d\tan x}{dx}={\sec }^{2}x}$.
	Explain why ${\displaystyle \frac{d\cot x}{dx}=-{\cot }^{2}x}$.
	Explain why ${\displaystyle \frac{d\sec x}{dx}=\tan x\sec x}$.
	Explain why ${\displaystyle \frac{d\csc x}{dx}=-\cot x\csc x}$.
	Explain why ${\displaystyle \frac{d\ln x}{dx}=\frac{1}{x}}$.
	Explain why ${\displaystyle \frac{d{\sin }^{-1}x}{dx}=\frac{1}{\sqrt{1-x^2}}}$.
	Explain why ${\displaystyle \frac{d{\cos }^{-1}x}{dx}=-\frac{1}{\sqrt{1-x^2}}}$.
	Explain why ${\displaystyle \frac{d{\tan }^{-1}}{dx}=\frac{1}{1+x^2}}$.
	Explain why ${\displaystyle \frac{d{\cot }^{-1}x}{dx}=-\frac{1}{1+x^2}}$.
	Explain why ${\displaystyle \frac{d{\csc }^{-1}x}{dx}=-\frac{1}{\left|x\right|\sqrt{x^2-1}}}$.

Computing some derivatives

	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\left(2x+3\right)\left(5x+6\right)}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\left(x+\frac{1}{x}\right)\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(2x-5\right)}^2{\left(3x-4\right)}^3}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\left(e{x}^2+\frac{\pi }{x^3}+x^{7/2}\right)}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\frac{x-3}{x-4}\right)}^2}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{3x+5}{4-x^2}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{x}{\sqrt{1-2x}}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{1+\sqrt{x}}{1-\sqrt{x}}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{2\left(x+1\right)}{x^2+2x-3}}$.
	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 ${\displaystyle y=\frac{x^2-2}{x-1}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{\sqrt{x}}{\sqrt{x-3}}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{x^n+1}{x^{\mathrm{n}}-1}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{\sqrt{1+x^2}}{\sqrt{1-x^2}}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{2x^2-1}{x\sqrt{1+x^2}}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=u^n}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sqrt{1-x^2}}$.

Correcting derivative identities

	Correct the identity ${\displaystyle \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}+u\frac{dv}{dx}}{v^2}}$
	Correct the identity ${\displaystyle \frac{d}{dx}\left(u+v\right)=\frac{du}{dv}-\frac{dv}{dx}}$
	Correct the identity ${\displaystyle \frac{d}{dx}\left(uv\right)=\frac{du}{dx}·\frac{dv}{dx}}$

Verifying derivative identities

	If ${\displaystyle y=x^{7/2}}$ show that ${\displaystyle 2x\frac{dy}{dx}-7y=0}$.
	If ${\displaystyle y=3-x^2}$ show that ${\displaystyle {\left(\frac{dy}{dx}\right)}^2-4x^2=0}$.
	If ${\displaystyle y=\sqrt{x}+\frac{1}{\sqrt{x}}}$ show that ${\displaystyle 2x\frac{dy}{dx}+y-\sqrt{x}=0}$.
	If ${\displaystyle y=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots +\frac{x^n}{n!}}$ show that ${\displaystyle \frac{dy}{dx}-y+\frac{x^n}{n!}=0}$.
	If ${\displaystyle y=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots }$ show that ${\displaystyle \frac{dy}{dx}-y=0}$.
	If ${\displaystyle z=\frac{3}{1+t}}$ show that ${\displaystyle 3t\frac{dz}{dt}=z\left(z-3\right)}$.
	If ${\displaystyle y=\frac{1}{a-\mathrm{z}}}$ show that ${\displaystyle \frac{dz}{dy}={\left(z-a\right)}^2}$.
	If ${\displaystyle y=\frac{x}{x-p}}$ show that ${\displaystyle x\frac{dy}{dx}=y\left(1-y\right)}$.
	If ${\displaystyle y=x-\sqrt{1+x^2}}$ show that ${\displaystyle \left(1+x^2\right){\left(\frac{dy}{dx}\right)}^2=y^2}$.
	If ${\displaystyle y=x^2}$ show that ${\displaystyle {\left(\frac{dy}{dx}\right)}^2=4y}$.
	If ${\displaystyle y=\sqrt{1+x^5}}$ show that ${\displaystyle \frac{dy}{dx}=\frac{5x^4}{2y}}$.

Derivatives at a point

	Find ${\displaystyle \frac{dy}{dx}}$ at $x=3$ when $y=x^6+3x^2+5$.
	Find ${\displaystyle {\left.\frac{dy}{dx}\right|}_{x=3}}$ when $y=\left(x+1\right)\left(x+2\right)$.

Derivatives with respect to functions

	Differentiate ${\displaystyle t^2-\frac{4}{t^4}}$ with respect to ${\displaystyle t^5}$.
	Differentiate ${\displaystyle \frac{x^2}{1+x^2}}$ with respect to ${\displaystyle x^2}$.
	Differentiate ${\displaystyle \frac{ax+b}{cx+d}}$ with respect to ${\displaystyle \frac{a\prime x+b\prime }{c\prime x+d\prime }}$.
	Differentiate ${\displaystyle x^3}$ with respect to ${\displaystyle x^2}$.
	Differentiate ${\displaystyle \frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}}$ with respect to ${\displaystyle \sqrt{1-x^4}}$.
	Differentiate ${\displaystyle \frac{x}{1+x^2}}$ with respect to ${\displaystyle x^3}$.
	Differentiate ${\displaystyle x-\sqrt{1-x^2}}$ with respect to ${\displaystyle \sqrt{1-x^2}}$.
	Differentiate ${\displaystyle 7x^5-11x^2}$ with respect to ${\displaystyle 7x^2-15\mathrm{x}}$.

Derivatives of parametric functions

	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x=pt}$ and ${\displaystyle y=p/t}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x=a{t}^2}$ and ${\displaystyle y=2at}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{2a{t}^2}{1+t^2}}$ and ${\displaystyle x=\frac{2a}{1+t^2}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle a\frac{1-t^2}{1+t^2}}$ and ${\displaystyle y=b\frac{2t}{1+t^2}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x=a\sqrt{\frac{t^2-1}{t^2+1}}}$ and ${\displaystyle y=at\sqrt{\frac{t^2-1}{t^2+1}}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x=a\frac{1-t^2}{1+t^2}}$ and ${\displaystyle y=\frac{2bt}{1-t^2}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x=\frac{3at}{1+t^3}}$ and ${\displaystyle y=\frac{3a{t}^2}{1+t^3}}$.
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle x=\frac{1-t^2}{1+t^2}}$ and ${\displaystyle y=\frac{2t}{1+t^2}}$.

Implicit differentiation

	Find ${\displaystyle \frac{dy}{dx}}$ when $x^4+y^4=4a^2{x}^2{y}^2.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when $x^5+y^5-5a{x}^2{y}^2=0.$
	If $a{x}^2+b{y}^2+2gx+2fy+2hxy+c=0$ show that ${\displaystyle \frac{dy}{dx}+\frac{ax+hy+g}{hx+by+f}=0.}$
	If $xy+px+q=0$ prove that ${\displaystyle x^2\frac{dy}{dx}}$ is always constant.
	Find ${\displaystyle \frac{dy}{dx}}$ when $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0.$

Derivatives with trigonometric functions.

	Find ${\displaystyle \frac{dy}{dx}}$ when $y=\sin \left(3x+2\right).$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y=\sqrt{\sin x^4}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y=x^2\sin x.$
	Find ${\displaystyle \frac{dy}{dx}}$ when $y=\tan x\sin 2x.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sin x^2-\frac{\tan x}{1+x^2}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{2\cos x-x}{x+2}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\left(1+x^2\right)+\frac{x}{\sin x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{\sin 2x}{\cos x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sin \left(\frac{x}{3}\right)\csc \left(\frac{2x}{3}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sin \left(\sin x+\cos x\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sqrt{{\sec }^{2}x+{\csc }^{2}x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\left(x^2-1\right)\left(\cot x+\frac{\tan x}{1+x^2}\right)}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sqrt{\frac{\cos \theta -\sin \theta }{\cos \theta +\sin \theta }}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{\sec x+\tan x}{\sec x-\tan x}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sqrt{\frac{1-\cos x}{1+\cos x}}}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x^3{\tan }^2\left(\frac{x}{2}\right)}.$
	If $y=\tan \left(\cos \left(\sin \theta \right)\right)$ find ${\displaystyle \frac{dy}{dx}}.$

Derivatives with exponentials and logs.

	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\left(e{x}^2+\frac{\pi }{x^3}+x^{\frac{7}{2}}\right).}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=a^{ax+b}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=a^{x^3}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=6^{2x}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\ln \left(a{x}^2+b\right).}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=e^{3\ln x}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=e^{2x}-e^{-2x}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=e^{x^2+2x}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=a^x{x}^a.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x{e}^x.}$

References [PLACEHOLDER]

[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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