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\(\dfrac{d}{dx}\ln x\)=_______.

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True or False \(\dfrac{d}{dx}x^{e}=e\,x^{e-1}.\)

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True or False \(\dfrac{d}{dx}\ln [x\sin ^{2}x ] =\frac{d}{dx} \ln x\cdot \dfrac{d}{dx}\ln \sin ^{2}x\).

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True or False \(\dfrac{d}{dx}\ln \pi =\dfrac{1}{\pi }\).

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\(\dfrac{d}{dx}\ln \vert x\vert =\) _______ for all \(x≠ 0\).

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\(\lim\limits_{n\rightarrow \infty }\left( 1+\dfrac{1}{n}\right) ^{n}=\) _______.

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\(y=5\ln x\)

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\(y=-3\ln x\)

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\(y=\log _{2}u\)

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\(y=\log _{3}u\)

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\(y=(\cos x)(\ln x)\)

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\(y=(\sin x)(\ln x)\)

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\(y=\ln (3x)\)

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\(y=\ln \dfrac{x}{2}\)

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\(y=\ln (e^{t}-e^{-t})\)

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\(y=\ln (e^{at}+e^{-at})\)

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\(y=x\ln (x^{2}+4)\)

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\(y=x\ln (x^{2}+5x+1)\)

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\(y=v\ln \sqrt{v^{2}+1}\)

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\(y=v\ln \sqrt[3]{3v+1}\)

229

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\(y=\dfrac{1}{2}\ln \dfrac{1+x}{1-x}\)

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\(y=\dfrac{1}{2}\ln \dfrac{1+x^{2}}{1-x^{2}}\)

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\(y=\ln (\ln x)\)

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\(y=\ln \left( \ln \dfrac{1}{x}\right)\)

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\(y=\ln \dfrac{x}{\sqrt{x^{2}+1}}\)

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\(y=\ln \dfrac{4x^{3}}{\sqrt{x^{2}+4}}\)

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\(y=\ln \dfrac{(x^{2}+1)^{2}}{x\sqrt{x^{2}-1}}\)

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\(y=\ln \dfrac{x\sqrt{3x-1}}{(x^{2}+1)^{3}}\)

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\(y=\ln (\sin \theta )\)

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\(y=\ln (\cos \theta )\)

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\(y=\ln (x+\sqrt{x^{2}+4})\)

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\(y=\ln (\sqrt{x+1}+\sqrt{x})\)

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\(y=\log _{2}(1+x^{2})\)

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\(y=\log _{2}(x^{2}-1)\)

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\(y=\tan^{-1}(\ln x)\)

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\(y=\sin ^{-1}(\ln x)\)

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\(y=\ln (\tan ^{-1}t)\)

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\(y=\ln (\sin ^{-1}t)\)

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\(y=(\ln x)^{1/2}\)

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\(y=(\ln x)^{-1/2}\)

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\(y=\sin (\ln \theta)\)

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\(y=\cos (\ln \theta)\)

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\(y=x \ln \sqrt{\cos (2x) }\)

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\(y=x^{2}\ln \sqrt{\sin (2x) }\)

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\(x\ln y+y\ln x=2\)

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\(\dfrac{\ln y}{x}+\dfrac{\ln x}{y}=2\)

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\(\ln (x^{2}+y^{2})=x+y\)

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\(\ln (x^{2}-y^{2})=x-y\)

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\(\ln \dfrac{y}{x} =y\)

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\(\ln \dfrac{y}{x}-\ln \dfrac{x}{y}=1\)

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\(y=(x^{2}+1)^{2}(2x^{3}-1)^{4}\)

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\(y=(3x^{2}+4)^{3}(x^{2}+1)^{4}\)

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\(y=\dfrac{x^{2}(x^{3}+1)}{\sqrt{x^{2}+1}}\)

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\(y=\dfrac{\sqrt{x}(x^{3}+2)^{2}}{\sqrt[3]{3x+4}}\)

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\(y=\dfrac{x\cos x}{(x^{2}+1)^{3}\sin x}\)

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\(y=\dfrac{x\sin x}{( 1+e^{x}) ^{3}\cos x}\)

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\(y=(3x)^{x}\)

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\(y=(x-1)^{x}\)

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\(y=x^{\ln x}\)

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\(y=(2x) ^{\ln x}\)

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\(y=x^{x^{2}}\)

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\(y=(3x)^{\sqrt{x}}\)

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\(y=x^{{e}^{{x}}}\)

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\(y=(x^{2}+1)^{{e}^{{x}}}\)

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\(y=x^{\sin x}\)

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\(y=x^{\cos x}\)

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\(y=(\sin x)^{x}\)

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\(y=(\cos x)^{x}\)

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\(y=(\sin x)^{\cos x}\)

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\(y=(\sin x)^{\tan x}\)

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\(x^{y}=4\)

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\(y^{x}=10\)

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\(y=\ln (5x)\) at \(\left(\dfrac{1}{5},0\right)\)

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\(y=x\ln x\) at \((1,0)\)

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\(y=\dfrac{x^{2}\sqrt{3x-2}}{( x-1) ^{2}}\) at \((2, 8)\)

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\(y=\dfrac{x (\sqrt[3]{x} +1) ^{2}}{\sqrt{x+1}}\) at \((8,24)\)

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\(\lim\limits_{n \rightarrow \infty}\left( 1+\dfrac{1}{n}\right) ^{2n}\)

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\(\lim\limits_{n \rightarrow \infty}\left(1+\dfrac{1}{n}\right)^{n/2}\)

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\(\lim\limits_{n \rightarrow \infty}\left(1+\dfrac{1}{3n}\right)^{n}\)

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\(\lim\limits_{n \rightarrow \infty}\left(1+\dfrac{4}{n}\right) ^{n}\)

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Find \(\dfrac{d^{10}}{dx^{10}}(x^{9}\ln x)\).

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If \(f(x)=\ln (x-1)\), find \(f^{(n)}(x)\).

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If \(y=\ln (x^{2}+y^{2})\), find the value of \(\dfrac{dy}{dx}\) at the point \((1,0)\).

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If \(f(x) =\tan \left( \ln x-\dfrac{1}{\ln x} \right)\), find \(f^\prime (e)\).

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Find \(y^\prime\) if \(y=x^{x},\) \(x > 0\), by using \(y=x^{x}=e^{\ln x^{x}}\) and the Chain Rule.

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If \(y=\ln (kx)\), where \(x > 0\) and \(k > 0\) is a constant, show that \(y^\prime =\dfrac{1}{x}.\)

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\(y=x\tan ^{-1}\dfrac{x}{a}-\dfrac{1}{2}a\ln (x^{2}+a^{2}),\quad a≠ 0\)

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\(y=x\sin ^{-1}\dfrac{x}{a}+a\ln \sqrt{a^{2}-x^{2}}\), \(\vert a \vert > \vert x\vert\), \(a≠ 0\)

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1. Show that if the annual rate of interest \(r\) is compounded continuously, then the amount \(A\) after \(t\) years is \(A={\it Pe}^{rt}\), where \(P\) is the initial investment.
2. If an initial investment of \(P=$5000\) earns 2% interest compounded continuously, how much is the investment worth after \(10\) years?
3. How long does it take an investment of $10,000 to double if it is invested at 2.4% compounded continuously?
4. Show that the rate of change of \(A\) with respect to \(t\) when the interest rate \(r\) is compounded continuously is \(\dfrac{dA}{dt}=rA.\)

230

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A bank offers a certificate of deposit (CD) that matures in \( 10\) years with a rate of interest of 3% compounded continuously. (See Problem 89.) Suppose you purchase such a CD for $2000 in your IRA.

1. Write an equation that gives the amount \(A\) in the CD as a function of time \(t\) in years.
2. How much is the CD worth at maturity?
3. What is the rate of change of the amount \(A\) at \(t=3?\) At \(t=5?\) At \( t=8?\)
4. Explain the results found in (c).

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Sound Level of a Leaf Blower The loudness \(L\), measured in decibels (dB), of a sound of intensity \(I\) is defined as \(L(x)=10\log \dfrac{ I(x)}{I_{0}}\), where \(x\) is the distance in meters from the source of the sound and \(I_{0}=10^{-12}\) W/m\(^{2}\) is the least intense sound that a human ear can detect. The intensity \(I\) is defined as the power \(P\) of the sound wave divided by the area \(A\) on which it falls. If the wave spreads out uniformly in all directions, that is, if it is spherical, the surface area is \(A(x)=4\pi x^{2}\)m\(^{2}\), and \(I(x)= \dfrac{P}{4\pi x^{2}}\)W/m\(^{2}\).

1. If you are 2.0 m from a noisy leaf blower and are walking away from it, at what rate is the loudness \(L\) changing with respect to distance \(x\)?
2. Interpret the sign of your answer.

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Show that \(\ln x+\ln y=2x\) is equivalent to \(xy=e^{2x}.\) Use this equation to find \(y^\prime\). Compare this result to the solution found in Example 1(c).

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If \(\ln T=kt\), where \(k\) is a constant, show that \(\dfrac{dT}{dt}=kT\).

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Graph \(y=\left( 1+\dfrac{1}{x}\right) ^{x}\) and \(y=e\) on the same set of axes. Explain how the graph supports the fact that \( \lim\limits_{n\rightarrow \infty }\left( 1+\dfrac{1}{n}\right) ^{n}=e\).

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Power Rule for Functions Show that if \(u\) is a function of \(x\) that is differentiable and \(a\) is a real number, then \[ \dfrac{d}{dx}[u( x) ] ^{a}=a[ u( x) ] ^{a-1}u^\prime (x) \] provided \(u^{a}\) and \(u^{a-1}\) are defined. [\(Hint:\) Let \(\vert y\vert =\vert [u( x)] ^{a}\vert\) and use logarithmic differentiation.]

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Show that the tangent lines to the graphs of the family of parabolas \(f( x) =-\dfrac{1}{2}x^{2}+k\) are always perpendicular to the tangent lines to the graphs of the family of natural logarithms \(g(x) =\ln (bx) +c\), where \(b > 0\), \(k\), and \(c\) are constants.

Source: Mathematics students at Millikin University, Decatur, Illinois.

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Show that \(2x-\ln (3+6e^{x}+3e^{2x})=C-2\ln (1+e^{-x})\) for some constant \(C\).

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If \(f\) and \(g\) are differentiable functions, and if \(f(x) > 0\), show that \[ \frac{d}{dx}f(x)^{g(x)}=g(x)f(x)^{g(x)-1}f^\prime ( x) +f( x) ^{g(x)}[\ln f(x)]g^\prime (x) \]