From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Mon, 2 Sep 2019 13:46:59 +0900
Subject: Initial commit

---
 .../latex/contrib/probsoln/samples/prob-easy2.tex  | 137 +++++++++++++++++++++
 1 file changed, 137 insertions(+)
 create mode 100644 macros/latex/contrib/probsoln/samples/prob-easy2.tex

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diff --git a/macros/latex/contrib/probsoln/samples/prob-easy2.tex b/macros/latex/contrib/probsoln/samples/prob-easy2.tex
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+ % This file is public domain
+ %
+ % These are all easy differentiation problems
+
+\begin{defproblem}{diffeasy:gpowh}%
+\begin{onlyproblem}%
+\(f(x) = g(x)^{h(x)}.\)%
+\end{onlyproblem}%
+\begin{onlysolution}%
+\begin{eqnarray*}
+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{eqnarray*}%
+\end{onlysolution}%
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:arcsin}%
+\begin{onlyproblem}%
+\(y = \arcsin(x)\)%
+\end{onlyproblem}%
+\begin{onlysolution}%
+\[\sin(y) = x\]
+diff. w.r.t. $x$:
+\begin{eqnarray*}
+\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{eqnarray*}
+\end{onlysolution}%
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:arccos}%
+\begin{onlyproblem}%
+$y = \arccos x$.%
+\end{onlyproblem}%
+\begin{onlysolution}%
+\(\cos y = x\)
+diff. w.r.t. $x$:
+\begin{eqnarray*}
+-\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{eqnarray*}
+\end{onlysolution}%
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:tan}%
+\begin{onlyproblem}%
+\(y = \tan x\)%
+\end{onlyproblem}
+\begin{onlysolution}%
+\begin{eqnarray*}
+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{eqnarray*}
+\end{onlysolution}%
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:arctan}%
+\begin{onlyproblem}%
+\(y = \arctan x = \tan^{-1}x\)%
+\end{onlyproblem}%
+\begin{onlysolution}%
+\[\tan y = x\]
+diff w.r.t. $x$:
+\begin{eqnarray*}
+\sec^2y\frac{dy}{dx} & = & 1\\
+\frac{dy}{dx} & = & \frac{1}{\sec^2y}\\
+ & = & \frac{1}{1+\tan^2y}\\
+ & = & \frac{1}{1+x^2}
+\end{eqnarray*}
+\end{onlysolution}%
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:cot}%
+\begin{onlyproblem}%
+\(y = (\tan x)^{-1} = \cot x\)%
+\end{onlyproblem}
+\begin{onlysolution}%
+\begin{eqnarray*}
+\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{eqnarray*}
+\end{onlysolution}%
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:cosxsqsinx}%
+\begin{onlyproblem}%
+$y = \cos(x^2)\sin x$.%
+\end{onlyproblem}%
+\begin{onlysolution}%
+\[\frac{dy}{dx} = -\sin(x^2)2x\sin x + \cos(x^2)\cos x\]
+\end{onlysolution}%
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:xlnx}%
+\begin{onlyproblem}%
+$y = (x+1)\ln(x+1)$.
+\end{onlyproblem}%
+\begin{onlysolution}%
+\begin{eqnarray*}
+\frac{dy}{dx} & = & \ln(x+1) + \frac{x+1}{x+1}\\
+ & = & 1 + \ln(x+1).
+\end{eqnarray*}
+\end{onlysolution}
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:glng}%
+\begin{onlyproblem}%
+$f(x) = g(x)\ln(g(x))$.
+\end{onlyproblem}
+\begin{onlysolution}%
+\begin{eqnarray*}
+f'(x) & = & g'(x)\ln(g(x)) + \frac{g(x)}{g(x)}g'(x)\\
+ & = & g'(x)(1+\ln(g(x))).
+\end{eqnarray*}
+\end{onlysolution}%
+\end{defproblem}
+
+\begin{defproblem}{diffeasy:sinx/x}
+\begin{onlyproblem}%
+$y = \frac{\sin x}{x}$.
+\end{onlyproblem}
+\begin{onlysolution}%
+\[\frac{dy}{dx} = \frac{\cos x}{x} - \frac{\sin x}{x^2}\]
+\end{onlysolution}%
+\end{defproblem}
+
-- 
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