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The Derivative

1
The Definition of the Derivative

Problem  1
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Mr. McKeague cc
Mr. McKeague
Problem  2

Use the definition of the derivative to find the derivative of \(f(x)=x^2\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Preston
Preston
David espanol spanish
David
Problem  3
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CJ
CJ
Gordon espanol spanish
Gordon
Problem  4

Use the definition of the derivative to find the derivative of \(f(x)=x^3\)

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Mr. McKeague cc
Mr. McKeague
Preston
Preston
Problem
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Mr. McKeague cc
Mr. McKeague
CJ
CJ
Gordon espanol spanish
Gordon
Problem  6
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Mr. McKeague cc
Mr. McKeague
CJ
CJ
Gordon espanol spanish
Gordon
Problem  7

Find the derivative of \(f(x)=\displaystyle\frac{1}{x}\)

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CJ
CJ
Problem  8

Use the definition of the derivative to find the derivative of \(f(x)=3\sqrt{x}\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Aaron
Aaron
David espanol spanish
David
Problem  9

Find the value of the derivative of \(f(x)=3\sqrt{x}\) when \(x=16\).

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Mr. McKeague cc
Mr. McKeague
CJ
CJ
Gordon espanol spanish
Gordon
Problem  10
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Molly S. cc
Molly S.
Preston
Preston
Problem  11
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Molly S. cc
Molly S.
Preston
Preston
David espanol spanish
David
Problem  12

AP Calculus Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague
Problem  13

AP Calculus Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague
David cc espanol spanish
David

2
Power and Sum Rules

Problem  1

Differentiate each function.

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

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

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

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

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Mr. McKeague cc
Mr. McKeague
Matt cc
Matt
Problem  2

Differentiate each function.

  1. \(y=(2x-3)^2\)

  2. \(y=\sqrt{x}(x-3)\)

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Preston
Preston
Katrina
Katrina
Problem  3

Use the power rule to find the derivative of each function.

  1. \(y=x^5\)

  2. \(f(x)=\displaystyle\frac{1}{x^3}\)

  3. \(g(x)=\sqrt{x}\)

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Mr. McKeague cc
Mr. McKeague
Preston
Preston
David espanol spanish
David
Problem  4

Use the power rule to find the derivative of each function.

  1. \(y=4x^5\)

  2. \(f(x)=\displaystyle\frac{3}{x^3}\)

  3. \(g(x)=4\sqrt{x}\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Preston
Preston
Problem  5

Find \(\displaystyle\frac{dy}{dx}\) for \(y=x^8+12x^5-4x^4+10x^3-6x+5\).

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Matt
Matt
Katrina
Katrina
Preston
Preston
David espanol spanish
David
Problem  6
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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  7

Find \(\displaystyle\frac{dy}{dx}\) for \(y=5x^3+\displaystyle\frac{8}{x}-5\sqrt[3]{x}\)

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Preston
Preston
Problem  8

Find the points on the curve \(y=x^4-6x^2+4\) where the tangent line is horizontal.

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Matt
Matt
Katrina
Katrina
Preston
Preston
David espanol spanish
David
Problem  9
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Molly S. cc
Molly S.
Gordon
Gordon
Gordon espanol spanish
Gordon
Problem  10
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Gordon
Gordon
Gordon espanol spanish
Gordon
Problem  11
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CJ
CJ
Problem  12

Suppose that, during a flu epidemic, the number of people ill with flu symptoms can be approximated by the function \[N(t)=90t^2-t^3, \quad 0\leq t \leq 70\] where \(t\) is measured in days since the beginning of the epidemic.

  1. At what rate are flu symptoms spreading on day \(15\) of the epidemic, and how many people are ill with the flu on day \(15\)?

  2. At what rate is the flu spreading on day \(65\)?

  3. When is the flu spreading at the rate of \(1500\) people per day?

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Mr. McKeague cc
Mr. McKeague

3
Product and Quotient Rules

Problem  1

Differentiate each function.

  1. \(G(x)=\left(x^2+x+1\right)\left(x^2+2\right)\)

  2. \(H(x)=\left(x^3-x+1\right)\left(x^{-2}+2x^{-3}\right)\)

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Mr. McKeague cc
Mr. McKeague
David espanol spanish
David
Preston
Preston
Problem  2

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

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
CJ cc
CJ
David cc espanol spanish
David
Problem  3
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Mr. McKeague cc
Mr. McKeague
Gordon
Gordon
Gordon espanol spanish
Gordon
Problem  4

Differentiate \(f(t)=\sqrt{t}(a+bt)\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Aaron
Aaron
David espanol spanish
David
Problem  5
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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  6

Differentiate each function.

  1. \(h(x)=\displaystyle\frac{x+1}{x-1}\)

  2. \(f(x)=\displaystyle\frac{x^5}{x^3-2}\)

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Mr. McKeague cc
Mr. McKeague
Preston
Preston
David espanol spanish
David
Problem  7
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Mr. McKeague cc
Mr. McKeague
Problem  8

Use the quotient rule to find the derivative of \(y=\displaystyle\frac{x^3-8}{x-2}\).

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Aaron
Aaron
David espanol spanish
David
Problem  9

Find \(f'(2)\) if \(f(x)=\displaystyle\frac{x^3+2x-1}{x^2-1}\)

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Mr. McKeague cc
Mr. McKeague
Aaron cc
Aaron
David cc espanol spanish
David
Problem  10

Differentiate \(g(x)=\displaystyle\frac{3x^2+2\sqrt{x}}{x}\)

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
Problem  11

Differentiate \(f(x)=\displaystyle\frac{x}{x+\displaystyle\frac{c}{x}}\)

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
David espanol spanish
David
Problem  12
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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon

4
Trigonometric Functions

Problem  1

Differentiate each function.

  1. \(y=\tan x\)

  2. \(y=x-3\sin x\)

  3. \(g(t)=t^3\cos t\)

  4. \(y=\sec{\theta}\tan{\theta}\)

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
Problem  2

Find the equation of the line tangent to \(y=x+\cos{x}\) at the point \((0,1)\).

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
Problem  3

Find an equation of the tangent line to the graph of \(f(x)=\sin x\) at \(x=\displaystyle\frac{4\pi}{3}\)

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Betsy
Betsy
Gordon espanol spanish
Gordon
Problem  4

Differentiate \(y=x^2\sin x\)

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Mr. McKeague cc
Mr. McKeague
Betsy
Betsy
Gordon espanol spanish
Gordon
Problem  5

Differentiate \(y=\cos^2x\)

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Mr. McKeague cc
Mr. McKeague
Betsy
Betsy
Molly S. cc
Molly S.
Problem  6

Find \(\displaystyle\frac{dy}{dx}\) for \(y=\displaystyle\frac{\sec{t}}{1+\sec{t}}\)

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

Find the second derivative of \(f(x)=\sec x\)

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

5
Chain Rule

Problem  1

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

  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  2

Differentiate each function.

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

  2. \(y=\tan{(\sin x)}\)

  3. \(y=\ln{(\sin t)}\)

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

  5. \(y=\ln{(e^{2t})}\)

  6. \(z=\tan{(e^{-3\theta})}\)

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
David espanol spanish
David
Problem  3
Choose instructor to watch:
Aaron
Aaron
Problem  4

Differentiate \(y=\left(4x^3+3x+1\right)^7\)

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Mr. McKeague cc
Mr. McKeague
Betsy
Betsy
CJ
CJ
Problem  5

Use the Power Rule to differentiate \(y=\displaystyle\frac{1}{\left(x^2+1\right)}\)

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Betsy
Betsy
CJ
CJ
Problem  6

Differentiate \(y=\displaystyle\frac{1}{\left(7x^5-x^4+2\right)^{10}}\)

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Betsy
Betsy
CJ
CJ
Problem  7

Differentiate \(y=\tan^3x\)

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Betsy
Betsy
CJ
CJ
Problem  8

Differentiate \(y=\displaystyle\frac{\left(x^2-1\right)^3}{(5x+1)^8}\)

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Betsy
Betsy
CJ
CJ
Problem  9

Differentiate \(y=\sqrt{\displaystyle\frac{2x-3}{8x+1}}\)

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Betsy
Betsy
CJ
CJ
Problem  10

Differentiate \(y=\cos 4x\)

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Betsy
Betsy
Problem  11

AP Calculus Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague

6
Implicit Differentiation

Problem  1
  1. Find \(\displaystyle\frac{dy}{dx}\) for \(x^2+y^2=16\).

  2. Find \(\displaystyle\frac{dy}{dx}\) for \(xy-x-3y-4=0\).

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

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Mr. McKeague cc
Mr. McKeague
Problem  2

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

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
Problem  3

Find \(\displaystyle\frac{dy}{dx}\) for \(\sqrt{xy}=1+x^2y\)

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
Problem  4

AP Calculus Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague

7
Exponential Functions

Problem  1

Find \(\displaystyle\frac{dy}{dx}\) for \(y=x^2e^x\).

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Preston
Preston
David espanol spanish
David
Problem  2

Find the equation of the line tangent to the curve \(y=2xe^x\) at \((0,0)\).

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David
David
Preston
Preston
David espanol spanish
David
Problem  3

Find \(f'(x)\) for \(f(x)=xe^x\csc{x}\).

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
Problem  4

Find \(\displaystyle\frac{dy}{dx}\) for \(e^{\frac{x}{y}}=x-y\).

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Mr. McKeague cc
Mr. McKeague
Aaron cc
Aaron
Problem  5
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Aaron
Aaron

8
Logarithmic Functions

Problem  1

Differentiate each function.

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

  2. \(y=\tan{(\sin x)}\)

  3. \(y=\ln{(\sin t)}\)

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

  5. \(y=\ln{(e^{2t})}\)

  6. \(z=\tan{(e^{-3\theta})}\)

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Mr. McKeague cc
Mr. McKeague
Aaron
Aaron
David espanol spanish
David
Problem  2

Differentiate each function.

  1. \(y=x^5\)

  2. \(y=e^x\)

  3. \(y=5^x\)

  4. \(y=x^x\)

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

  6. \(y=\left(\ln{x}\right)^x\)

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Mr. McKeague cc
Mr. McKeague