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Math 112 Solutions for 2.10, p. 206

1.
(b) $ \dfrac{x}{2y}$. (c) $ \dfrac{x-y}{x+y}$. (f) $ \dfrac{y-y^{4}}{3x+y^{4}}$. (l) $ \dfrac{x/y}{\vert x/y\vert}$.
2.
(a) $ -\dfrac{y^{2}+x^{2}}{y^{3}}=\dfrac{-1}{y^{3}}$. (c) $ \dfrac{\sqrt y+\sqrt x}{2x\sqrt x}=\dfrac{1}{2x\sqrt x}$.
3.
(c) $ \vert x\vert+\dfrac{x^{2}}{\vert x\vert}$. (f) $ \dfrac{2x}{x^{2}+4}$. (g) $ 2x\ln\vert x\vert+x$. (h) $ \cot x$. (j) $ \dfrac{\sec x\tan x+\sec^{2}x}{\sec x+\tan x}=\sec x$. (n) $ \dfrac{e^{\sin^{-1}x}}{\sqrt{1-x^{2}}}+\dfrac{e^{x}}{\sqrt{1-e^{2x}}}$.
4.
$ \sin^{-1}x$ and $ \cos^{-1}x$ are the two acute angles in the same right triangle, so their sum is $ \dfrac{\pi}{2}$. $ \dfrac{d}{dx}(\cos^{-1}x)=\dfrac{-1}{\sqrt{1-x^{2}}}$.
7.
(a) $ \frac{d}{dx}(\ln\vert x\vert)=\frac{1}{\vert x\vert}\cdot\frac{x}{\vert x\vert}
=\frac{x}{x^{2}}=\frac{1}{x}$. (b) $ \frac{d}{dx}\ln
f(x)=\frac{1}{f(x)}f'(x)=\frac{f'(x)}{f(x)}$. (c) $ \frac{d}{dx}\ln\vert f(x)\vert=
\frac{1}{\vert f(x)\vert}\cdot
\frac{f(x)f'(x)}{\vert f(x)\vert}=\frac{f(x)f'(x)}{f(x)^{2}}=\frac{f'(x)}{f(x)}$.
8.
(c) $ y\left(\dfrac{12x^{2}}{x^{3}+1}+\dfrac{12x^{3}}{x^{4}+1}-\dfrac{5}{x-1}-
\dfrac{6}{x-2}\right)$.





Jason Grout 2003-02-12