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- % 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}
-