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Applied Calculus
Derivatives of Exponential and Logarithmic Functions

1
The Exponential Functions

Mini Lecture
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Exponential functions and logarithmic functions are very special functions, and so are their derivatives.

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Mr. McKeague cc
Mr. McKeague
Mini Lecture

As your ability in mathematics increases, the definitions become more abstract and complicated. The number e is an important number and you need to understand its definition.

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Mr. McKeague cc
Mr. McKeague
Problem  3
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Use tables to graph \(f(x)=2^x\) and \(g(x)=\left(\dfrac{1}{2}\right)^x\)

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Stephanie
Stephanie
Octabio cc
Octabio
Winston
Winston
Octabio cc espanol spanish
Octabio
Problem  4
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Graph: \(f(t)=e^{0.3t}\); \(f(t)=e^{1t}\); \(f(t)=e^{2t}\)

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Octabio cc
Octabio
Stephanie
Stephanie
Octabio cc espanol spanish
Octabio
Problem  5
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Graph: \(f(t)=e^{-0.3t}\); \(f(t)=e^{-1t}\); \(f(t)=e^{-2t}\)

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Stephanie
Stephanie
Octabio cc
Octabio
Winston
Winston
Octabio cc espanol spanish
Octabio
Problem  6
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Using the exponential function \(A(t)=5e^{0.06931t}\) predict how many bacteria will be on the counter after \(2\) hours.

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Stephanie
Stephanie
Octabio cc
Octabio
Winston
Winston
Octabio cc espanol spanish
Octabio
Problem  7
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If in a discussion group of \(10\) people, the first-ranked person participated \(35\) times, use the exponential function \[N(p)=N_1e^{-0.11(p-1)}\quad 1\leq p\leq 10\] to determine how many times the sixth-ranked person participates.

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Stephanie
Stephanie
Octabio cc
Octabio
Winston
Winston
Octabio cc espanol spanish
Octabio
Problem  8
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Use the functionA(t)=P\left(1+\frac{r}{n}\right)^{nt}A(t)=P(1+nr)ntto determine the amount of money accumulated after 1515 years if \$2\text{,}000$2,000 is invested in an account that pays 8\%8% interest compounded quarterly.

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Octabio cc
Octabio
Winston
Winston
Octabio cc espanol spanish
Octabio

2
The Natural Logarithm Function

Study Skill
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Logarithmic functions are very closely related to exponential functions, both in their definitions and their inverse functions.

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Mr. McKeague cc
Mr. McKeague
Problem  2
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Convert each exponential function to the corresponding logarithmic function.

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

  2. \(a=e^{4x+3}\)

  3. \(20=e^{-0.3x}\)

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Octabio cc
Octabio
Breylor
Breylor
Octabio cc espanol spanish
Octabio
Problem  3
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Convert each logarithmic function to the corresponding exponential function.

  1. \(5=\ln{(7x)}\)

  2. \(x+5=\ln{y-3}\)

  3. \(-0.01=\ln{x+8}\)

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Octabio cc
Octabio
Breylor
Breylor
Octabio cc espanol spanish
Octabio
Study Skill
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Natural logarithms, two special identities, and a postage stamp.

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Mr. McKeague cc
Mr. McKeague
Problem  5
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Evaluate each of the following:

  1. \(\ln e\)

  2. \(\ln 1\)

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Octabio cc
Octabio
Breylor
Breylor
Octabio cc espanol spanish
Octabio
Problem  6
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Use the properties of the natural logarithm to expand each logarithmic expression.

  1. \(\ln(3x)\)

  2. \(\ln\left(\dfrac{8x}{x+4}\right)\)

  3. \(\ln x^6\)

  4. \(12\text{,}000\ln\left(xy^5\right)\)

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Joshua cc
Joshua
Breylor
Breylor
Problem  7
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Use the properties of the natural logarithm to write each logarithmic expression as an expression with a single logarithm.

  1. \(\ln x+6\ln y\)

  2. \(\ln a-\ln b -\ln c\)

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Joshua cc
Joshua
Problem  8
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To four decimal places, approximate the solution to the equation \(e^{2x+5}=12\).

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Joshua cc
Joshua

3
Differentiating the Natural Logarithm Function

Mini Lecture
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Justifying the derivative for the natural logarithm function. An intuitive look, then the actual derivation.

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Mr. McKeague cc
Mr. McKeague
Problem  2
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For the function \(f(x)=7x^3\ln{x}\), find

  1. \(f'(x)\)

  2. an equation of the line tangent at \(x=e\).

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Mr. McKeague cc
Mr. McKeague
Mr. Schwennicke cc
Mr. Schwennicke
Julieta cc
Julieta
Problem  3
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Find \(f'(x)\) for \(f(x)=\ln{\left(5x^2+2x-7\right)}\).

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Mr. McKeague cc
Mr. McKeague
Mr. Schwennicke cc
Mr. Schwennicke
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  4
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Find \(f'(x)\) for \(f(x)=\ln\left(x^2-4\right)^3\)

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Mr. McKeague cc
Mr. McKeague
Logan cc
Logan
Saba cc
Saba
Problem  5
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Find \(f'(x)\) for \(f(x)=[\ln\left(3x\right)]^4\)

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Mr. McKeague cc
Mr. McKeague
Joshua cc
Joshua
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  6
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Find \(f'(x)\) for \(f(x)=\dfrac{6}{\ln\left(8x^2\right)}\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Logan cc
Logan
Problem  7
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Find \(y'\) for \(3x^4y^2+\ln\left(xy^2\right)=6\)

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Mr. McKeague
Mr. McKeague
Joshua cc
Joshua

4
Differentiating the Natural Exponential Function

Mini Lecture
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Explaining the derivative of the natural exponential function.

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Mr. McKeague cc
Mr. McKeague
Problem  2
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Find \(f'(x)\) for \(f(x)=8x^3e^x\).

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Mr. McKeague cc
Mr. McKeague
Mr. Schwennicke cc
Mr. Schwennicke
Joshua cc
Joshua
Problem  3
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Find the equation of the line tangent to the curve \(f(x)=7xe^x\) at the point \((0,0)\). Round decimals to two places.

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Mr. McKeague cc
Mr. McKeague
Nathan
Nathan
Saba cc
Saba
Problem  4
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Find \(f'(x)\) for \(f(x)=e^{3x^2+5x}\).

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Mr. McKeague cc
Mr. McKeague
Bailey cc
Bailey
Joshua cc
Joshua
Mr. Schwennicke cc
Mr. Schwennicke
Take 5
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A little more about natural exponential functions, their derivatives, and equations of tangent lines.

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Mr. McKeague cc
Mr. McKeague
Problem  6
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Find \(f'(3)\) for \(f(x)=\ln\left(2x+e^{-0.05x}\right)\). Round the result to \(4\) decimal places.

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Nathan
Nathan
Saba cc
Saba
Problem  7
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Find \(f''(x)\) for \(f(x)=e^{2x^2+5}\)

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Nathan
Nathan
Saba cc
Saba
Problem  8
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Find \(f'(x)\) for \(f(x)=6e^{-7x}\ln 4x\)

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Saba cc
Saba
Joshua cc
Joshua
Problem  9
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Find \(f'(0)\) for \(f(x)=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}\)

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Saba cc
Saba
Joshua cc
Joshua
Problem  10
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Find the derivative.

  1. \(\dfrac{d}{dx}\left[e^{4x}\right]\)

  2. \(\dfrac{d}{dx}\left[e^{-6x}\right]\)

  3. \(\dfrac{d}{dx}\left[5e^{-0.2x}\right]\)

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Logan cc
Logan
Saba cc
Saba