% 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}