\DTLifdbexists{problems}%
{\PackageError{datatool}{Database `problems' 
already exists}{}%
\aftergroup\endinput}{}% 
\bgroup\makeatletter
\dtl@message{Reconstructing database 
`problems'}%
\expandafter\global\expandafter
\newtoks\csname dtlkeys@problems\endcsname
\expandafter\global
 \csname dtlkeys@problems\endcsname={%
\db@plist@elt@w \db@col@id@w 1\db@col@id@end@ \db@key@id@w Label\db@key@id@end@ \db@type@id@w 0\db@type@id@end@ \db@header@id@w Label\db@header@id@end@ \db@col@id@w 1\db@col@id@end@ \db@plist@elt@end@ \db@plist@elt@w \db@col@id@w 2\db@col@id@end@ \db@key@id@w Question\db@key@id@end@ \db@type@id@w 0\db@type@id@end@ \db@header@id@w Question\db@header@id@end@ \db@col@id@w 2\db@col@id@end@ \db@plist@elt@end@ \db@plist@elt@w \db@col@id@w 3\db@col@id@end@ \db@key@id@w Answer\db@key@id@end@ \db@type@id@w 0\db@type@id@end@ \db@header@id@w Answer\db@header@id@end@ \db@col@id@w 3\db@col@id@end@ \db@plist@elt@end@ \db@plist@elt@w \db@col@id@w 4\db@col@id@end@ \db@key@id@w Year\db@key@id@end@ \db@type@id@w 1\db@type@id@end@ \db@header@id@w Year\db@header@id@end@ \db@col@id@w 4\db@col@id@end@ \db@plist@elt@end@ }
\expandafter\global\expandafter
\newtoks\csname dtldb@problems\endcsname
\expandafter\global
\csname dtldb@problems\endcsname={%
\db@row@elt@w \db@row@id@w 1\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w tan\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = \tan x$\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \begin {align*} y & = \tan x\\ & = \frac {\sin x}{\cos x}\\ \frac {dy}{dx} & = \frac {\cos x}{\cos x} + \sin x\times \frac {-1}{\cos ^2x}\times -\sin x\\ & = 1 + \tan ^2x\\ & = & \sec ^2x. \end {align*}\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 2013\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 1\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 2\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w cosxsqsinx\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = \cos (x^2)\sin x$.\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \[\frac {dy}{dx} = -\sin (x^2)2x\sin x + \cos (x^2)\cos x\]\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 2013\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 2\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 3\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w exp3x+2\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = \exp (3x+2)$ \db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \[\frac {dy}{dx} = 3\exp (3x+2)\]\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 2013\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 3\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 4\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w cubic\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y=x^3 + 4x^2 - x + 3$ \db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \[\frac {dy}{dx} = 3x^2 + 8x - 1\]\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 2013\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 4\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 5\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w xlnx\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = (x+1)\ln (x+1)$.\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \begin {align*} \frac {dy}{dx} & = \ln (x+1) + \frac {x+1}{x+1}\\ & = 1 + \ln (x+1). \end {align*}\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 2013\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 5\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 6\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w arccos\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = \arccos x$.\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \(\cos y = x\) diff. w.r.t. $x$: \begin {align*} -\sin y \frac {dy}{dx} & = 1\\ \frac {dy}{dx} & = \frac {-1}{\sin y}\\ & = \frac {-1}{\sqrt {1-\cos ^2y}}\\ & = \frac {-1}{\sqrt {1-x^2}} \end {align*}\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 2013\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 6\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 7\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w quad\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y=2x^3 + 6x -1$ \db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \[\frac {dy}{dx} = 6x + 6 = 6(x+1)\]\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 0\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 7\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 8\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w sinx/x\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = \frac {\sin x}{x}$.\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \[\frac {dy}{dx} = \frac {\cos x}{x} - \frac {\sin x}{x^2}\]\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 0\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 8\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 9\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w gpowh\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $f(x) = g(x)^{h(x)}.$\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \begin {align*} f(x) & = e^{\ln g(x)^{h(x)}}\\ & = e^{h(x)\ln g(x)}\\ f'(x) & = e^{h(x)\ln g(x)}(h'(x)\ln g(x) + h(x)\frac {g'(x)}{g(x)})\\ & = g(x)^{h(x)}(h'(x)\ln g(x) + \frac {h(x)g'(x)}{g(x)}) \end {align*}\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 0\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 9\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 10\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w arctan\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = \arctan x = \tan ^{-1}x$\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \[\tan y = x\] diff w.r.t. $x$: \begin {align*} \sec ^2y\frac {dy}{dx} & = 1\\ \frac {dy}{dx} & = \frac {1}{\sec ^2y}\\ & = \frac {1}{1+\tan ^2y}\\ & = \frac {1}{1+x^2} \end {align*}\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 0\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 10\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 11\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w arcsin\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = \arcsin (x)$\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \[\sin (y) = x\] diff. w.r.t. $x$: \begin {align*} \cos y \frac {dy}{dx} & = 1\\ \frac {dy}{dx} & = \frac {1}{\cos y}\\ & = \frac {1}{\sqrt {1 - \sin ^2y}}\\ & = \frac {1}{\sqrt {1-x^2}}. \end {align*}\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 0\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 11\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 12\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w exp4x\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = \exp (4x)$ \db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \[\frac {dy}{dx} = 4\exp (4x)\]\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 0\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 12\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 13\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w glng\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $f(x) = g(x)\ln (g(x))$.\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \begin {align*} f'(x) & = g'(x)\ln (g(x)) + \frac {g(x)}{g(x)}g'(x)\\ & = g'(x)(1+\ln (g(x))). \end {align*}\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 0\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 13\db@row@id@end@ \db@row@elt@end@ \db@row@elt@w \db@row@id@w 14\db@row@id@end@ \db@col@id@w 1\db@col@id@end@ \db@col@elt@w cot\db@col@elt@end@ \db@col@id@w 1\db@col@id@end@ \db@col@id@w 2\db@col@id@end@ \db@col@elt@w $y = (\tan x)^{-1} = \cot x$\db@col@elt@end@ \db@col@id@w 2\db@col@id@end@ \db@col@id@w 3\db@col@id@end@ \db@col@elt@w \begin {align*} \frac {dy}{dx} & = -(\tan x)^{-2}\sec ^2x\\ & = -\frac {\cos ^2x}{\sin ^2x}\cdot \frac {1}{\cos ^2x}\\ & = \frac {-1}{\sin ^2x}\\ & = -\csc ^2x. \end {align*}\db@col@elt@end@ \db@col@id@w 3\db@col@id@end@ \db@col@id@w 4\db@col@id@end@ \db@col@elt@w 0\db@col@elt@end@ \db@col@id@w 4\db@col@id@end@ \db@row@id@w 14\db@row@id@end@ \db@row@elt@end@ }
\expandafter\global
 \expandafter\newcount\csname dtlrows@problems\endcsname
\expandafter\global
 \csname dtlrows@problems\endcsname=14\relax
\expandafter\global
 \expandafter\newcount\csname dtlcols@problems\endcsname
\expandafter\global
 \csname dtlcols@problems\endcsname=4\relax
\expandafter
 \gdef\csname dtl@ci@problems@Label\endcsname{1}% 
\expandafter
 \gdef\csname dtl@ci@problems@Question\endcsname{2}% 
\expandafter
 \gdef\csname dtl@ci@problems@Answer\endcsname{3}% 
\expandafter
 \gdef\csname dtl@ci@problems@Year\endcsname{4}% 
\egroup
\def\dtllastloadeddb{problems}