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Math 112 Solutions for 2.9, p. 195

2.
$ \dfrac{dG}{dW}$ is the rate of change of population growth rate as a function of average per capita wage, in people per thousand per dollars per year. $ \dfrac{dQ}{dP}$ is the rate of change of average per capita wage as a function of mine production rate, measured in dollars per year per tons per year. $ \dfrac{dG}{dP}$ is the rate of change of population growth rate as a function of mine production rate, measured in people per thousand per tons per year.
3.
(b) $ 5(2x-x^{2})^{4}(2-2x)$. (c) $ -\frac{8}{x^{2}}(\frac{2}{x})^{3}$. (d) $ 3(x-\frac{1}{x})^{2}(1+\frac{1}{x^{2}})$. (f) $ 3(1-\cos x)^{2}\sin x$. (g) $ -2x\sin(x^{2})$. (h) $ 5\sec^{2}5x$. (j) $ 2xe^{x^{2}}$. (k) $ 3\cdot 10^{3x}\ln 10$. (l) $ 2^{\sin
x}\ln 2\cos x$.
4.
(b) $ 3\sec 3x\tan 3x\tan 4x+4\sec 3x\sec^{2}4x$. (d) $ -5\pi
\cos^{4}\pi x\sin \pi x$. (f) $ e^{-x}(-\cos 3x-3\sin 3x)$. (l) $ \dfrac{7e^{-.02x}}{(1+7e^{-.02x})^{2}}$.
5.
(c) $ -2x\sin(x^{2}+4)$. (f) $ 9(x^{3}-1)^{2}x^{2}$.
6.
(a) (iv)
7.
If $ y=[f(x)]^{n}$, let $ y=u^{n}, u=f(x)$. Then $ \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=nu^{n-1}f\,'(x)
=n[f(x)]^{n-1}f\,'(x)$.
8.
$ \frac{d}{dx}(a^{x})=\frac{d}{dx}(e^{x\ln a})=(\ln a)e^{x\ln
a}=(\ln a)a^{x}$.
9.
-42
11.
0
15.
$ a=\frac{dv}{dt}=\frac{dv}{ds}\cdot\frac{ds}{dt}=v\frac{dv}{ds}$.
18.
If $ f$ is an even function, then $ f(-x)=f(x)$. Differentiating, we get $ f\;'(-x)(-1)=f\;'(x)\Rightarrow f\;'(-x)=-f\;'(x)$, and $ f\;'$ is an odd function.





Jason Grout 2003-02-11