Calculus/Differentiation/Basics of Differentiation/Solutions

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Find the Derivative by Definition[edit| edit source]

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Prove the Constant Rule[edit| edit source]

10. Use the definition of the derivative to prove that for any fixed real number ,

Find the Derivative by Rules[edit| edit source]

Power Rule[edit| edit source]

11.
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Product Rule[edit| edit source]

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Quotient Rule[edit| edit source]

29.
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Chain Rule[edit| edit source]

43.
Let . Then
Let . Then
44.
Let . Then
Let . Then
45.
Let . Then
Let . Then
46.
Let . Then




Let . Then




47.
Let . Then




Let . Then




48.
Let . Then




Let . Then




49.
Let . Then




Let . Then




50.
Let . Then




Let . Then




51.
Let . Then




Let . Then




52.
Let . Then




Let . Then




53.
Let . Then
Let . Then

Exponentials[edit| edit source]

54.
55.
Let . Then
Let . Then
56.
Let

Then

Using the chain rule, we have

The individual factor are

So

Let

Then

Using the chain rule, we have

The individual factor are

So

57.

Logarithms[edit| edit source]

58.
47.
59.
Let . Then
Let . Then
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Trigonometric functions[edit| edit source]

62.
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More Differentiation[edit| edit source]

65.
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67.
Let . Then



Let . Then



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Implicit Differentiation[edit| edit source]

Use implicit differentiation to find y'

74.




75.








Logarithmic Differentiation[edit| edit source]

Use logarithmic differentiation to find :

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80.




Equation of Tangent Line[edit| edit source]

For each function, , (a) determine for what values of the tangent line to is horizontal and (b) find an equation of the tangent line to at the given point.

81.
:

a)
b)


:

a)
b)


82.
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b)


:

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83.
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84.
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85.
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86.
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/ b)


:

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/ b)


87. Find an equation of the tangent line to the graph defined by at the point (1,-1).