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3.3 Assess Your Understanding

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Concepts and Vocabulary

  1. ddxlnx=_______.

1x

  1. True or False ddxxe=exe−1.

True

  1. True or False ddxln[xsin2x]=ddxlnx⋅ddxlnsin2x.

False

  1. True or False ddxlnπ=1π.

False

  1. ddxln|x|= _______ for all x≠0.

1x

  1. lim _______.

e

Skill Building

In Problems 7–44, find y^\prime.

  1. y=5\ln x

y'=5/x

  1. y=-3\ln x

  1. y=\log _{2}u

y'={1\over u \ \ln 2}

  1. y=\log _{3}u

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

y'={1\over x}\cos x- \ \sin x \ \ln x

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

  1. y=\ln (3x)

y'=1/x

  1. y=\ln \dfrac{x}{2}

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

y'={e^t+e^{-t}\over e^t-e^{-t}}

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

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

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

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

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

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

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

229

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

y'={1\over 1-x^2}

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

  1. y=\ln (\ln x)

y'={1\over x \ \ln x}

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

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

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

  1. y=\ln \dfrac{4x^{3}}{\sqrt{x^{2}+4}}

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

y'=\dfrac{4x}{x^2+1} -\dfrac{1}{x} - \dfrac{x}{x^2-1}

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

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

y'=\cot\theta

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

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

y'={1\over \sqrt{x^2+4}}

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

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

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

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

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

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

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

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

y'={1\over \tan^{-1} t}\cdot {1\over 1+t^2}

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

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

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

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

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

y'=\cos(\ln\theta)\cdot{1\over \theta}

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

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

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

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

In Problems 45–50, use implicit differentiation to find y^\prime =\dfrac{dy}{dx}.

  1. x\ln y+y\ln x=2

{dy\over dx}=- \dfrac{y^2 +xy \ \ln y}{x^2 +xy \ \ln x}

  1. \dfrac{\ln y}{x}+\dfrac{\ln x}{y}=2

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

{dy\over dx}={x^2+y^2-2x\over 2y-x^2-y^2}

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

  1. \ln \dfrac{y}{x} =y

{dy\over dx}={y\over x(1-y)}

  1. \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.

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

y'=(x^2+1)^2(2x^3-1)^4\left({4x\over x^2+1}+{24x^2\over 2x^3-1}\right)

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

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

y'={x^2(x^3+1)\over \sqrt{x^2+1}} \left({2\over x}+{3x^2\over x^3+1}-{x\over x^2+1}\right)

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

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

y'={x \ \cos x\over (x^2+1)^3 \ \sin x}\left({1\over x}- \ \tan x-{6x\over x^2+1}-\ \cot x\right)

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

  1. y=(3x)^{x}

y'=(3x)^x(\ln(3x)+1)

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

  1. y=x^{\ln x}

y'=2x^{\ln x}{\ln x\over x}

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

  1. y=x^{x^{2}}

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

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

  1. y=x^{{e}^{{x}}}

y'=x^{e^x}\left(e^x\, \ln\, x+{e^x\over x}\right)

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

  1. y=x^{\sin x}

y'=x^{\sin x}\left(\cos \,x\,\ln x+{\sin x\over x}\right)

  1. y=x^{\cos x}

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

y'=(\sin x)^x\left(\ln(\sin x)+x\cot x\right)

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

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

y'=(\sin x)^{\cos x}\left(-\sin x\, \ln(\sin x)+\,\cos x\,\cot x\right)

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

  1. x^{y}=4

y'={-y\over x\,\ln x}

  1. 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.

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

y=5 x-1

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

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

y=-5 x+18

  1. 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.

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

e^2

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

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

e^{1/3}

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

Applications and Extensions

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

{d^{10}y\over dx^{10}}=\frac{362880}{x}

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

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

{dy\over dx}=2

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

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

y'=x^x(\ln\, x+1)

  1. 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.

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

y'=\tan^{-1}(x/a)

  1. 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.

    1. (a) 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. (b) If an initial investment of P=$5000 earns 2% interest compounded continuously, how much is the investment worth after 10 years?
    3. (c) How long does it take an investment of $10,000 to double if it is invested at 2.4% compounded continuously?
    4. (d) 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.

  1. (a) See Student Solutions Manual.
  2. (b) \$6107.01
  3. (c) 28.881 years
  4. (d) See Student Solutions Manual.

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

  1. 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. (a) 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. (b) Interpret the sign of your answer.

  1. (a) -{10\over \ln 10} \,\rm{dB}/{m}
  2. (b) Answers will vary.

  1. 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).

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

See Student Solutions Manual.

  1. 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.

  1. 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.]

See Student Solutions Manual.

  1. 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

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

See Student Solutions Manual.

  1. 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)