Concepts and Vocabulary
\(\dfrac{d}{dx}\ln x\)=_______.
True or False \(\dfrac{d}{dx}x^{e}=e\,x^{e-1}.\)
True or False \(\dfrac{d}{dx}\ln [x\sin ^{2}x ] =\frac{d}{dx} \ln x\cdot \dfrac{d}{dx}\ln \sin ^{2}x\).
True or False \(\dfrac{d}{dx}\ln \pi =\dfrac{1}{\pi }\).
\(\dfrac{d}{dx}\ln \vert x\vert =\) _______ for all \(x≠ 0\).
\(\lim\limits_{n\rightarrow \infty }\left( 1+\dfrac{1}{n}\right) ^{n}=\) _______.
Skill Building
In Problems 7–44, find \(y^\prime\).
\(y=5\ln x\)
\(y=-3\ln x\)
\(y=\log _{2}u\)
\(y=\log _{3}u\)
\(y=(\cos x)(\ln x)\)
\(y=(\sin x)(\ln x)\)
\(y=\ln (3x)\)
\(y=\ln \dfrac{x}{2}\)
\(y=\ln (e^{t}-e^{-t})\)
\(y=\ln (e^{at}+e^{-at})\)
\(y=x\ln (x^{2}+4)\)
\(y=x\ln (x^{2}+5x+1)\)
\(y=v\ln \sqrt{v^{2}+1}\)
\(y=v\ln \sqrt[3]{3v+1}\)
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\(y=\dfrac{1}{2}\ln \dfrac{1+x}{1-x}\)
\(y=\dfrac{1}{2}\ln \dfrac{1+x^{2}}{1-x^{2}}\)
\(y=\ln (\ln x)\)
\(y=\ln \left( \ln \dfrac{1}{x}\right)\)
\(y=\ln \dfrac{x}{\sqrt{x^{2}+1}}\)
\(y=\ln \dfrac{4x^{3}}{\sqrt{x^{2}+4}}\)
\(y=\ln \dfrac{(x^{2}+1)^{2}}{x\sqrt{x^{2}-1}}\)
\(y=\ln \dfrac{x\sqrt{3x-1}}{(x^{2}+1)^{3}}\)
\(y=\ln (\sin \theta )\)
\(y=\ln (\cos \theta )\)
\(y=\ln (x+\sqrt{x^{2}+4})\)
\(y=\ln (\sqrt{x+1}+\sqrt{x})\)
\(y=\log _{2}(1+x^{2})\)
\(y=\log _{2}(x^{2}-1)\)
\(y=\tan^{-1}(\ln x)\)
\(y=\sin ^{-1}(\ln x)\)
\(y=\ln (\tan ^{-1}t)\)
\(y=\ln (\sin ^{-1}t)\)
\(y=(\ln x)^{1/2}\)
\(y=(\ln x)^{-1/2}\)
\(y=\sin (\ln \theta)\)
\(y=\cos (\ln \theta)\)
\(y=x \ln \sqrt{\cos (2x) }\)
\(y=x^{2}\ln \sqrt{\sin (2x) }\)
In Problems 45–50, use implicit differentiation to find \(y^\prime =\dfrac{dy}{dx}.\)
\(x\ln y+y\ln x=2\)
\(\dfrac{\ln y}{x}+\dfrac{\ln x}{y}=2\)
\(\ln (x^{2}+y^{2})=x+y\)
\(\ln (x^{2}-y^{2})=x-y\)
\(\ln \dfrac{y}{x} =y\)
\(\ln \dfrac{y}{x}-\ln \dfrac{x}{y}=1\)
In Problems 51–72, use logarithmic differentiation to find \(y^\prime\). Assume that the variable is restricted so that all arguments of logarithm functions are positive.
\(y=(x^{2}+1)^{2}(2x^{3}-1)^{4}\)
\(y=(3x^{2}+4)^{3}(x^{2}+1)^{4}\)
\(y=\dfrac{x^{2}(x^{3}+1)}{\sqrt{x^{2}+1}}\)
\(y=\dfrac{\sqrt{x}(x^{3}+2)^{2}}{\sqrt[3]{3x+4}}\)
\(y=\dfrac{x\cos x}{(x^{2}+1)^{3}\sin x}\)
\(y=\dfrac{x\sin x}{( 1+e^{x}) ^{3}\cos x}\)
\(y=(3x)^{x}\)
\(y=(x-1)^{x}\)
\(y=x^{\ln x}\)
\(y=(2x) ^{\ln x}\)
\(y=x^{x^{2}}\)
\(y=(3x)^{\sqrt{x}}\)
\(y=x^{{e}^{{x}}}\)
\(y=(x^{2}+1)^{{e}^{{x}}}\)
\(y=x^{\sin x}\)
\(y=x^{\cos x}\)
\(y=(\sin x)^{x}\)
\(y=(\cos x)^{x}\)
\(y=(\sin x)^{\cos x}\)
\(y=(\sin x)^{\tan x}\)
\(x^{y}=4\)
\(y^{x}=10\)
In Problems 73–76, find an equation of the tangent line to the graph of \(y=f(x)\) at the given point.
\(y=\ln (5x)\) at \(\left(\dfrac{1}{5},0\right)\)
\(y=x\ln x\) at \((1,0)\)
\(y=\dfrac{x^{2}\sqrt{3x-2}}{( x-1) ^{2}}\) at \((2, 8)\)
\(y=\dfrac{x (\sqrt[3]{x} +1) ^{2}}{\sqrt{x+1}}\) at \((8,24)\)
In Problems 77–80, express each limit in terms of \(e.\)
\(\lim\limits_{n \rightarrow \infty}\left( 1+\dfrac{1}{n}\right) ^{2n}\)
\(\lim\limits_{n \rightarrow \infty}\left(1+\dfrac{1}{n}\right)^{n/2}\)
\(\lim\limits_{n \rightarrow \infty}\left(1+\dfrac{1}{3n}\right)^{n}\)
\(\lim\limits_{n \rightarrow \infty}\left(1+\dfrac{4}{n}\right) ^{n}\)
Applications and Extensions
Find \(\dfrac{d^{10}}{dx^{10}}(x^{9}\ln x)\).
If \(f(x)=\ln (x-1)\), find \(f^{(n)}(x)\).
If \(y=\ln (x^{2}+y^{2})\), find the value of \(\dfrac{dy}{dx}\) at the point \((1,0)\).
If \(f(x) =\tan \left( \ln x-\dfrac{1}{\ln x} \right)\), find \(f^\prime (e)\).
Find \(y^\prime\) if \(y=x^{x},\) \(x > 0\), by using \(y=x^{x}=e^{\ln x^{x}}\) and the Chain Rule.
If \(y=\ln (kx)\), where \(x > 0\) and \(k > 0\) is a constant, show that \(y^\prime =\dfrac{1}{x}.\)
In Problems 87 and 88, find \(y^\prime\). Assume that \(a\) is a constant.
\(y=x\tan ^{-1}\dfrac{x}{a}-\dfrac{1}{2}a\ln (x^{2}+a^{2}),\quad a≠ 0\)
\(y=x\sin ^{-1}\dfrac{x}{a}+a\ln \sqrt{a^{2}-x^{2}}\), \(\vert a \vert > \vert x\vert\), \(a≠ 0\)
Continuously Compounded Interest In Problems 89 and 90, use the following discussion:
Suppose an initial investment, called the principal \(P\), earns an annual rate of interest \(r\), which is compounded \(n\) times per year. The interest earned on the principal \(P\) in the first compounding period is \(P\left( \dfrac{r}{n}\right)\), and the resulting amount \(A\) of the investment after one compounding period is \(A=P+P \left(\dfrac{r}{n}\right) =P\left( 1+\dfrac{r}{n}\right)\). After \(k\) compounding periods, the amount \(A\) of the investment is \(A=P\left(1+\dfrac{r}{n}\right)^{k}\). Since in \(t\) years there are \(nt\) compounding periods, the amount \(A\) after \(t\) years is \[ A=P\left( 1+\dfrac{r}{n}\right)^{nt} \] When interest is compounded so that after \(t\) years the accumulated amount is \(A=\) \(\lim\limits_{n\rightarrow \infty }P\left( 1+\dfrac{r}{n}\right) ^{nt}\), the interest is said to be compounded continuously.
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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.
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\(^{2}\).
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).
If \(\ln T=kt\), where \(k\) is a constant, show that \(\dfrac{dT}{dt}=kT\).
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 \( \lim\limits_{n\rightarrow \infty }\left( 1+\dfrac{1}{n}\right) ^{n}=e\).
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.]
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.
Challenge Problems
Show that \(2x-\ln (3+6e^{x}+3e^{2x})=C-2\ln (1+e^{-x})\) for some constant \(C\).
If \(f\) and \(g\) are differentiable functions, and if \(f(x) > 0\), show that \[ \frac{d}{dx}f(x)^{g(x)}=g(x)f(x)^{g(x)-1}f^\prime ( x) +f( x) ^{g(x)}[\ln f(x)]g^\prime (x) \]