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Applied Calculus
Differentiation: The Language of Change

1
The Derivative of a Function and Two Interpretations

Study Skill

Introduction and Definitions

  1. What is the definition of the derivative?

  2. If a function represents distance as a function of time, what does its derivative represent?

  3. If a function represents the cost to produce \(x\) items, what does its derivative cost?

  4. If the volume of a sphere is a function of its radius, what is the relationship between the rate of change of the volume and the rate of change of the radius?

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Mr. McKeague cc
Mr. McKeague
Problem  2

Find the derivative of the function \(f(x)=5x+8\).

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Mr. Damarest cc
Mr. Damarest
Logan cc
Logan
Lauren cc
Lauren
Problem  3
practice icon

If \(f(x)=x^2+6x-2\), find each of the following and interpret each result.

  1. \(f(5)\)

  2. \(f'(x)\)

  3. \(f'(5)\)

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Mr. McKeague cc
Mr. McKeague
Gordon cc
Gordon
Richard cc
Richard
Cynthia cc espanol spanish
Cynthia
Problem  4
practice icon
  1. If \(f(x)=4\), find \(f'(x)\)

  2. If \(f(x)=-16\), find \(f'(2)\)

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Mr. McKeague cc
Mr. McKeague
Gordon cc
Gordon
Joshua cc
Joshua
Cynthia cc espanol spanish
Cynthia
Problem  5
practice icon
  1. If \(y=x^2\), find \(\dfrac{dy}{dx}\)

  2. If \(f(x)=x^5\), find \(f'(x)\)

  3. If \(y=x=x^1\), find \(\dfrac{dy}{dx}\)

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Mr. Schwennicke cc
Mr. Schwennicke
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  6
practice icon

If \(f(x)=x^\frac{2}{3}\) find

  1. \(f(8)\)

  2. \(f'(8)\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Joshua cc
Joshua
Cynthia cc espanol spanish
Cynthia
Problem  7

If \(f(x)=\displaystyle\frac{1}{x^4}\), find

  1. \(f'(x)\)

  2. The equation of the line tangent at \((-1,1)\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Molly S. cc
Molly S.
Cynthia cc espanol spanish
Cynthia
Problem  8
practice icon

If \(f(x)=\displaystyle\frac{4}{x^8}\), find \(f'(x)\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Lauren cc
Lauren
Octabio cc espanol spanish
Octabio
Problem  9

If \(y=cx\), where \(c\) is a constant, find \(\displaystyle\frac{dx}{dy}\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Lauren cc
Lauren
Cynthia cc espanol spanish
Cynthia
Problem  10

Differentiate.

  1. \(y=8x\)

  2. \(f(x)=-14x\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Lauren cc
Lauren
Cynthia cc espanol spanish
Cynthia
Problem  11
practice icon

Find \(\displaystyle\frac{dy}{dx}\) for \(y=5x^3+4x^2+9x-7\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Richard cc
Richard
Cynthia cc espanol spanish
Cynthia
Problem  12
practice icon

If \(f(x)=x^2-4x+3\), find

  1. the value of \(x\) for which \(f(x)=0\).

  2. the value of \(x\) for which \(f'(x)=0\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Richard cc
Richard
Cynthia cc espanol spanish
Cynthia
Problem  13
practice icon

If \(y=9\sqrt[3]{x^2}\), find \(\displaystyle\frac{dy}{dx}\Bigg|_{x=8} \)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Richard cc
Richard
Cynthia cc espanol spanish
Cynthia
Problem  14

If \(f(x)=6x^4\), find \(\dfrac{dy}{dx}\)

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Mr. McKeague cc
Mr. McKeague
Lauren cc
Lauren
Betsy cc
Betsy
Cynthia
Cynthia
Problem  15
practice icon

Find the equation of a tangent line to the graph of \(f(x)=3x^4 +2x^3 -7x\) at the point \(x=-1\).

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  16

AP Calculus Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague
Problem  17

AP Calculus Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague
David cc espanol spanish
David
Problem  18

Differentiate each function.

  1. \(f(x)=x^2+3x-4\)

  2. \(V(r)=\dfrac{4}{3}\pi r^3\)

  3. \(F(x)=(16x)^3\)

  4. \(Y(t)=6t^{-9}\)

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Mr. McKeague cc
Mr. McKeague
Matt cc
Matt

2
Differentiating Products and Quotients

Problem  1
practice icon

Use the product rule to find the derivative of \(f(x)=\left(5x^2+4\right)\left(x^3+11\right)\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
CJ cc
CJ
David cc espanol spanish
David
Problem  2
practice icon

For \(g(t)=(t^2-4)(t^2+1)\), find

  1. the value of \(t\) for which \(g(t)=0\).

  2. the value of \(t\) for which \(g'(t)=0\).

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Gordon
Gordon
Betsy cc
Betsy
Joshua cc
Joshua
Cynthia cc espanol spanish
Cynthia
Problem  3
practice icon

Find \(f'(2)\) for \(f(x)=\dfrac{x^3+2x-1}{x^2-1}\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Aaron cc
Aaron
David cc espanol spanish
David
Problem  4

What information is contained in the following quotients?

  1. \(\dfrac{R_A(3)}{R_B(3)}\approx 1.003\)

  2. \(\dfrac{R'_A(3)}{R'_B(3)}=\dfrac{4623}{4097}\approx 1.13\)

  3. \(\left[\dfrac{R_A(t)}{R_B(t)}\right]'\approx 1.001\) when \(t=3\).

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Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  5
practice icon

Find the equation of the line tangent to the graph of \(f(x)=\displaystyle\frac{2x-5}{x-3}\) when \(x=4\).

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Gordon cc
Gordon
Betsy cc
Betsy
Richard cc
Richard
Octabio cc espanol spanish
Octabio
Problem  6

Using the function \[f(x)=\frac{\text{Number of votes}}{\text{Cost of those votes}}=\frac{2.3x}{7.1x^2+210}\]

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Joshua cc
Joshua
Betsy cc
Betsy
Richard cc
Richard
Octabio cc espanol spanish
Octabio
Problem  7

Find the equation of the tangent line to the graph of \(y=\left( 1+\sqrt{x}\right)\left(x-2\right)\) at \(x=4\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  8

Differentiate \(y=\dfrac{\left( x^2+1\right)\left(2x^2+1\right)}{3x^2+1}\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon

3
Higher Order Derivatives

Problem  1
practice icon

Find the first four derivatives for \(f(x)=3x^5-2x^4+x^2-10x+4\).

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Gordon cc
Gordon
Betsy cc
Betsy
Richard cc
Richard
Cynthia cc espanol spanish
Cynthia
Problem  2
practice icon

For the function \(f(x)=\dfrac{1}{2}x^4-4x^2\), find

  1. \(f(2)\)

  2. \(f'(2)\)

  3. \(f''(2)\)

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Gordon cc
Gordon
Betsy
Betsy
Joshua cc
Joshua
Cynthia cc espanol spanish
Cynthia
Problem  3
practice icon

If \(g(t)=t^3-9t\), find the values of \(t\) for which

  1. \(g(t)=0\)

  2. \(g'(t)=0\)

  3. \(g''(t)=0\)

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Gordon cc
Gordon
Betsy cc
Betsy
Richard cc
Richard
Cynthia cc espanol spanish
Cynthia
Problem  4
practice icon

If \(y=\dfrac{1}{x}\), find \(y'\), \(y''\), and \(y'''\).

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Gordon cc
Gordon
Betsy cc
Betsy
Richard cc
Richard
Cynthia cc espanol spanish
Cynthia

4
The Chain Rule and General Power Rule

Problem  1
practice icon

If \(y=u^2+3u\) and \(u=2x-9\), find

  1. \(\displaystyle\frac{dy}{dx}\)

  2. \(\displaystyle\frac{dy}{dx}\) when \(x=5\)

Interpret both results.

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Betsy cc
Betsy
Octabio cc espanol spanish
Octabio
Problem  2
practice icon

Find the derivatives of the following functions.

  1. \(y=\left(4x^3+5x+2\right)^6\)

  2. \(f(x)=\displaystyle\frac{4}{(6-2x)^5}\)

  3. \(y=\sqrt[5]{(7x-8)^3}\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Betsy cc
Betsy
Octabio cc espanol spanish
Octabio
Problem  3

Find the derivative of \(f(x)=\displaystyle\frac{(2x-1)^4}{(3x+2)}\) at \(x=-1\) and interpret this result.

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Mr. McKeague cc
Mr. McKeague
Lauren cc
Lauren
Richard cc
Richard
Gordon cc espanol spanish
Gordon
Problem  4

Find the derivative of \(f(x)=x^5(4x-1)^{1/4}\).

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Mr. McKeague cc
Mr. McKeague
Lauren cc
Lauren
Richard cc
Richard
Gordon cc espanol spanish
Gordon
Problem  5
practice icon

Find \(f'(x)\) for the function \(f(x)=\left(\displaystyle\frac{2x+5}{8x+7}\right)^4\).

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Mr. McKeague cc
Mr. McKeague
Lauren cc
Lauren
Richard cc
Richard
Gordon cc espanol spanish
Gordon
Mini Lecture

Differentiate each function

  1. \(y=\left(5x^3+4x^2\right)^5\)

  2. \(y=x^5\)

  3. \(y=\left(x^2+1\right)^{100}\)

  4. \(\sqrt{4+3x}\)

  5. \(y=\displaystyle\frac{1}{\left(t^4+1\right)^3}\)

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Mr. McKeague cc
Mr. McKeague
Katrina cc
Katrina
Aaron cc
Aaron
David cc espanol spanish
David
Problem  7

AP Calculus Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague

5
Implicit Differentiation

Problem  1
practice icon

Find and interpret \(y'\) for \(3x^5+2y^4+y=37\) at the point \((1,2)\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Joshua cc
Joshua
Octabio cc espanol spanish
Octabio
Problem  2
practice icon

Find \(y'\) for \(x^2y^3-3x+4y=10\).

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  3

Suppose both \(y\) and \(x\) are differentiable functions of \(t\) and that the relationship between \(y\) and \(x\) is expressed by the equation \(4x^3+3y^5=960\). Find and interpret \(\displaystyle\frac{dy}{dt}\) when \(\displaystyle\frac{dx}{dt}=4\), \(x=6\), and \(y=2\).

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Mr. Schwennicke cc
Mr. Schwennicke
Betsy cc
Betsy
Richard cc
Richard
Octabio cc espanol spanish
Octabio
Problem  4

Find the rate at which the area is increasing at the time that the radius of the spill is \(600\) feet.

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Betsy cc
Betsy
Joshua cc
Joshua
Octabio cc espanol spanish
Octabio
Problem  5

Use the function \(x=\displaystyle\frac{20000}{\sqrt[3]{2p^2-5}}+350\) to find the rate at which the number of instruments sold is changing with respect to time, when the price of an instrument is \(\$400\) and is changing at a rate of \(\$1\) per month.

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Joshua cc
Joshua
Betsy cc
Betsy
Richard cc
Richard
Octabio cc espanol spanish
Octabio
Problem  6
  1. Find \(\displaystyle\frac{dy}{dx}\) for \(x^2+y^2=16\).

  2. Find \(\displaystyle\frac{dy}{dx}\) for \(xy-x-3y-3=0\).

  3. Find the slope of the line tangent to \(y^3=x^3(2-x)\) at \((1,-1)\)

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Mr. McKeague cc
Mr. McKeague
Problem  7

AP Calculus Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague