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+ % This file is public domain
+ %
+ % These are all easy differentiation problems
+
+\newproblem{diffeasy:gpowh}{%
+\(f(x) = g(x)^{h(x)}.\)}{%
+\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*}}
+
+\newproblem{diffeasy:arcsin}{%
+\(y = \arcsin(x)\)}{%
+\[\sin(y) = x\]
+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*}}
+
+\newproblem{diffeasy:arccos}{%
+$y = \arccos x$.}{%
+\(\cos y = x\)
+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*}}
+
+\newproblem{diffeasy:tan}{%
+\(y = \tan x\)}{%
+\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*}}
+
+\newproblem{diffeasy:arctan}{%
+\(y = \arctan x = \tan^{-1}x\)}{%
+\[\tan y = x\]
+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*}}
+
+\newproblem{diffeasy:cot}{%
+\(y = (\tan x)^{-1} = \cot x\)}{%
+\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*}}
+
+\newproblem{diffeasy:cosxsqsinx}{%
+$y = \cos(x^2)\sin x$.}{%
+\[\frac{dy}{dx} = -\sin(x^2)2x\sin x + \cos(x^2)\cos x\]}
+
+\newproblem{diffeasy:xlnx}{%
+$y = (x+1)\ln(x+1)$.}{%
+\begin{eqnarray*}
+\frac{dy}{dx} & = & \ln(x+1) + \frac{x+1}{x+1}\\
+ & = & 1 + \ln(x+1).
+\end{eqnarray*}}
+
+\newproblem{diffeasy:glng}{%
+$f(x) = g(x)\ln(g(x))$.}{%
+\begin{eqnarray*}
+f'(x) & = & g'(x)\ln(g(x)) + \frac{g(x)}{g(x)}g'(x)\\
+ & = & g'(x)(1+\ln(g(x))).
+\end{eqnarray*}}
+
+\newproblem{diffeasy:sinx/x}{%
+$y = \frac{\sin x}{x}$.}{%
+\[\frac{dy}{dx} = \frac{\cos x}{x} - \frac{\sin x}{x^2}\]}
+
+\newproblem{diffeasy:exp4x}{%
+ $y = \exp(4x)$
+}%
+{%
+ \[\frac{dy}{dx} = 4\exp(4x)\]
+}
+
+\newproblem{diffeasy:exp3x+2}{%
+ $y = \exp(3x+2)$
+}%
+{%
+ \[\frac{dy}{dx} = 3\exp(3x+2)\]
+}
+
+\newproblem{diffeasy:cubic}{%
+ $y=x^3 + 4x^2 - x + 3$
+}%
+{%
+ \[\frac{dy}{dx} = 3x^2 + 8x - 1\]
+}
+
+\newproblem{diffeasy:quad}{%
+ $y=2x^3 + 6x -1$
+}%
+{%
+ \[\frac{dy}{dx} = 6x + 6 = 6(x+1)\]
+}