% Created by datatooltk on Thu Jun 26 15:43:38 BST 2014
\DTLifdbexists{mth102}%
{\PackageError{datatool}{Database `mth102'
already exists}{}%
\aftergroup\endinput}{}%
\bgroup\makeatletter
\dtl@message{Reconstructing database
`mth102'}%
\expandafter\global\expandafter
\newtoks\csname dtlkeys@mth102\endcsname
\expandafter\global
 \csname dtlkeys@mth102\endcsname={%
%
% header block for column 1
\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@ %
% header block for column 2
\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@ %
% header block for column 3
\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@ %
% header block for column 4
\db@plist@elt@w %
\db@col@id@w 4%
\db@col@id@end@ %
\db@key@id@w Level%
\db@key@id@end@ %
\db@type@id@w 1%
\db@type@id@end@ %
\db@header@id@w Level%
\db@header@id@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
\db@plist@elt@end@ %
% header block for column 5
\db@plist@elt@w %
\db@col@id@w 5%
\db@col@id@end@ %
\db@key@id@w Topic%
\db@key@id@end@ %
\db@type@id@w 0%
\db@type@id@end@ %
\db@header@id@w Topic%
\db@header@id@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
\db@plist@elt@end@ %
}%
\expandafter\global\expandafter
\newtoks\csname dtldb@mth102\endcsname
\expandafter\global
\csname dtldb@mth102\endcsname={%
%
% Start of row 1
\db@row@elt@w %
\db@row@id@w 1%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:gpowh%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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)}\left(h'(x)\ln g(x) + h(x)\frac {g'(x)}{g(x)}\right)\\
 & = g(x)^{h(x)}\left(h'(x)\ln g(x) + \frac {h(x)g'(x)}{g(x)}\right)
\end{align*}%
\db@col@elt@end@ %
\db@col@id@w 3%
\db@col@id@end@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 3%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 1
\db@row@id@w 1%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 2
\db@row@elt@w %
\db@row@id@w 2%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:arcsin%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 2%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 2
\db@row@id@w 2%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 3
\db@row@elt@w %
\db@row@id@w 3%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:arccos%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 2%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 3
\db@row@id@w 3%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 4
\db@row@elt@w %
\db@row@id@w 4%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:tan%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 2%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 4
\db@row@id@w 4%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 5
\db@row@elt@w %
\db@row@id@w 5%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:arctan%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 3%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 5
\db@row@id@w 5%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 6
\db@row@elt@w %
\db@row@id@w 6%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:cot%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 2%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 6
\db@row@id@w 6%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 7
\db@row@elt@w %
\db@row@id@w 7%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:cosxsqsinx%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 2%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 7
\db@row@id@w 7%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 8
\db@row@elt@w %
\db@row@id@w 8%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:xlnx%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 1%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 8
\db@row@id@w 8%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 9
\db@row@elt@w %
\db@row@id@w 9%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:glng%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 1%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 9
\db@row@id@w 9%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 10
\db@row@elt@w %
\db@row@id@w 10%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w diff:sinx/x%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\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@ %
% Column 3
\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@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 1%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Basic%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 10
\db@row@id@w 10%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 11
\db@row@elt@w %
\db@row@id@w 11%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w dfp:xcube%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\db@col@id@w 2%
\db@col@id@end@ %
\db@col@elt@w Differentiate $f(x) = x^3$ with respect to $x$ by first principles.%
\db@col@elt@end@ %
\db@col@id@w 2%
\db@col@id@end@ %
% Column 3
\db@col@id@w 3%
\db@col@id@end@ %
\db@col@elt@w \begin {align*}
 \frac {dy}{dx} & = \lim _{\Delta x\rightarrow 0}\frac {f(x+\Delta x) - f(x)}{\Delta x}\\ & = \lim _{\Delta x\rightarrow 0}\frac {(x+\Delta x)^3-x^3}{\Delta x}\\ & = \lim _{\Delta x\rightarrow 0}\frac {(x+\Delta x)(x^2+2x\Delta x+(\Delta x)^2)-x^3}{\Delta x}\\ & = \lim _{\Delta x\rightarrow 0}\frac {x^3+3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3-x^3}{\Delta x}\\ & = \lim _{\Delta x\rightarrow 0}\frac {3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3}{\Delta x}\\ & = \lim _{\Delta x\rightarrow 0}3x^2+3x\Delta x + (\Delta x)^2\\ & = 3x^2
\end {align*}%
\db@col@elt@end@ %
\db@col@id@w 3%
\db@col@id@end@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 3%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Theory%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 11
\db@row@id@w 11%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 12
\db@row@elt@w %
\db@row@id@w 12%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w dfp:Ioverxsq%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\db@col@id@w 2%
\db@col@id@end@ %
\db@col@elt@w Differentiate $\displaystyle f(x) = \frac {1}{x^2}$ with respect to $x$ by first principles.%
\db@col@elt@end@ %
\db@col@id@w 2%
\db@col@id@end@ %
% Column 3
\db@col@id@w 3%
\db@col@id@end@ %
\db@col@elt@w \begin {align*}
 \frac {df}{dx} & = \lim _{\Delta x\rightarrow 0}\frac {\frac {1}{(x+\Delta x)^2}-\frac {1}{x^2}}{\Delta x}\\ & = \lim _{\Delta x\rightarrow 0}\frac {\frac {x^2-(x+\Delta x)^2}{x^2(x+\Delta x)^2}}{\Delta x}\\ & = \lim _{\Delta x\rightarrow 0}\frac {x^2-(x^2+2x\Delta x+(\Delta x)^2)}{x^2\Delta x(x+\Delta x)^2}\\ & = \lim _{\Delta x\rightarrow 0}\frac {-2x\Delta x-(\Delta x)^2}{x^2\Delta x(x+\Delta x)^2}\\ & = \lim _{\Delta x\rightarrow 0}\frac {-2x-\Delta x}{x^2(x+\Delta x)^2}\\ & = \frac {-2x}{x^2x^2}\\ & = -\frac {2}{x^3} 
\end {align*}%
\db@col@elt@end@ %
\db@col@id@w 3%
\db@col@id@end@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 3%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Theory%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 12
\db@row@id@w 12%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 13
\db@row@elt@w %
\db@row@id@w 13%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w dfp:sqrtx%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\db@col@id@w 2%
\db@col@id@end@ %
\db@col@elt@w Differentiate from first principles $f(x) = \surd x$%
\db@col@elt@end@ %
\db@col@id@w 2%
\db@col@id@end@ %
% Column 3
\db@col@id@w 3%
\db@col@id@end@ %
\db@col@elt@w \begin {align*}
 \frac {df}{dx} & = \lim _{\Delta x\rightarrow 0}\frac {\sqrt {x+\Delta x}-\surd x}{\Delta x}\\ & = \lim _{\Delta x\rightarrow 0}\frac {(\sqrt {x+\Delta x}-\surd x)(\sqrt {x+\delta x}+\surd x)}{\Delta x(\sqrt {x+\Delta x}+\surd x)}\\ & = \lim _{\Delta x\rightarrow 0}\frac {x+\Delta x - x}{\Delta x(\sqrt {x+\Delta x}+\surd x)}\\ & = \lim _{\Delta x\rightarrow 0}\frac {\Delta x}{\Delta x(\sqrt {x+\Delta x}+\Delta x)}\\ & = \lim _{\Delta x\rightarrow 0}\frac {1}{\sqrt {x+\Delta x}+\surd x}\\ & = \frac {1}{2\surd x}
 \end {align*}%
\db@col@elt@end@ %
\db@col@id@w 3%
\db@col@id@end@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 3%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Theory%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 13
\db@row@id@w 13%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 14
\db@row@elt@w %
\db@row@id@w 14%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w dfp:cons%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\db@col@id@w 2%
\db@col@id@end@ %
\db@col@elt@w Differentiate from first principles $f(x) = c$ where $c$ is a constant.%
\db@col@elt@end@ %
\db@col@id@w 2%
\db@col@id@end@ %
% Column 3
\db@col@id@w 3%
\db@col@id@end@ %
\db@col@elt@w \begin {align*}
 \frac {df}{dx} & = \lim _{\Delta x\rightarrow 0}\frac {c-c}{\Delta x}\\
 & = \lim _{\Delta x\rightarrow 0}0\\ 
& = 0 
\end {align*}%
\db@col@elt@end@ %
\db@col@id@w 3%
\db@col@id@end@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 3%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Theory%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 14
\db@row@id@w 14%
\db@row@id@end@ %
\db@row@elt@end@ %
% Start of row 15
\db@row@elt@w %
\db@row@id@w 15%
\db@row@id@end@ %
% Column 1
\db@col@id@w 1%
\db@col@id@end@ %
\db@col@elt@w dfp:cosx%
\db@col@elt@end@ %
\db@col@id@w 1%
\db@col@id@end@ %
% Column 2
\db@col@id@w 2%
\db@col@id@end@ %
\db@col@elt@w Given \begin {align*} \lim _{x \rightarrow 0} \frac {\cos x - 1}{x} & = 0\\ \lim _{x \rightarrow 0} \frac {\sin x}{x} & = 1 \end {align*} differentiate from first principles $f(x) = \cos x$.%
\db@col@elt@end@ %
\db@col@id@w 2%
\db@col@id@end@ %
% Column 3
\db@col@id@w 3%
\db@col@id@end@ %
\db@col@elt@w \begin {align*}
 \frac {df}{dx} & = \lim _{\Delta x \rightarrow 0}\frac {f(x + \Delta x) - f(x)}{\Delta x}\\
 & = \lim _{\Delta x \rightarrow 0} \frac {\cos (x + \Delta x) - \cos (x)}{\Delta x}\\
 & = \lim _{\Delta x \rightarrow 0} \frac {\cos x\cos \Delta x - \sin x\sin \Delta x - \cos x}{\Delta x}\\
 & = \lim _{\Delta x \rightarrow 0} \frac {\cos x(\cos \Delta x - 1) - \sin x\sin \Delta x}{\Delta x}\\ & = \cos x\lim _{\Delta x \rightarrow 0}\frac {\cos \Delta x - 1}{\Delta x} - \sin x\lim _{\Delta x \rightarrow 0}\frac {\sin \Delta x}{\Delta x}\\
 & = -1 \qquad \text {(using given results)}
\end {align*}%
\db@col@elt@end@ %
\db@col@id@w 3%
\db@col@id@end@ %
% Column 4
\db@col@id@w 4%
\db@col@id@end@ %
\db@col@elt@w 3%
\db@col@elt@end@ %
\db@col@id@w 4%
\db@col@id@end@ %
% Column 5
\db@col@id@w 5%
\db@col@id@end@ %
\db@col@elt@w Theory%
\db@col@elt@end@ %
\db@col@id@w 5%
\db@col@id@end@ %
% End of row 15
\db@row@id@w 15%
\db@row@id@end@ %
\db@row@elt@end@ %
}%
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