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Calculus/Differentiation/Basics of Differentiation/Solutions

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Find the Derivative by Definition

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Prove the Constant Rule

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

Find the Derivative by Rules

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Power Rule

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Product Rule

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Quotient Rule

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Chain Rule

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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

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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

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59.
Let . Then
Let . Then
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Trigonometric functions

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More Differentiation

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67.
Let . Then



Let . Then



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Implicit Differentiation

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Use implicit differentiation to find y'

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Logarithmic Differentiation

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Use logarithmic differentiation to find :

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Equation of Tangent Line

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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.
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:

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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).










88. Find an equation of the tangent line to the graph defined by at the point (1,0).










Higher Order Derivatives

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89. What is the second derivative of ?




90. Use induction to prove that the (n+1)th derivative of a n-th order polynomial is 0.
base case: Consider the zeroth-order polynomial, .

induction step: Suppose that the n-th derivative of a (n-1)th order polynomial is 0. Consider the n-th order polynomial, . We can write where is a (n-1)th polynomial.

base case: Consider the zeroth-order polynomial, .

induction step: Suppose that the n-th derivative of a (n-1)th order polynomial is 0. Consider the n-th order polynomial, . We can write where is a (n-1)th polynomial.

Advanced Understanding of Derivatives

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91. Let be the derivative of . Prove the derivative of is .
Suppose . Let .


Therefore, if is the derivative of , then is the derivative of .
Suppose . Let .


Therefore, if is the derivative of , then is the derivative of .
92. Suppose a continuous function has three roots on the interval of . If , then what is ONE true guarantee of using
(a) the Intermediate Value Theorem;
(b) Rolle's Theorem;
(c) the Extreme Value Theorem.
These are examples only. More valid solutions may exist.
(a) is continuous. Ergo, the intermediate value theorem applies. There exists some such that , where .
(b) Rolle's Theorem does not apply for a non-differentiable function.
(c) is continuous. Ergo, the extreme value theorem applies. There exists a so that for all .
These are examples only. More valid solutions may exist.
(a) is continuous. Ergo, the intermediate value theorem applies. There exists some such that , where .
(b) Rolle's Theorem does not apply for a non-differentiable function.
(c) is continuous. Ergo, the extreme value theorem applies. There exists a so that for all .
93. Let , where is the inverse of . Let be differentiable. What is ? Else, why can not be determined?
If , then . We can use implicit differentiation.
If , then . We can use implicit differentiation.
94. Let where is a constant.

Find a value, if possible, for that allows each of the following to be true. If not possible, prove that it cannot be done.

(a) The function is continuous but non-differentiable.
(b) The function is both continuous and differentiable.
(a) .
. However, for , we find that , so makes the function continuous but non-differentiable.

(b) There is no that allows the function to be differentiable and continuous.

A proof of this is simple.
However,
To allow the best possible chance, we will let :
For any other , one will have an infinity on the left-hand sided limit. Therefore, there is no possible that allows the function to be differentiable and continuous.
(a) .
. However, for , we find that , so makes the function continuous but non-differentiable.

(b) There is no that allows the function to be differentiable and continuous.

A proof of this is simple.
However,
To allow the best possible chance, we will let :
For any other , one will have an infinity on the left-hand sided limit. Therefore, there is no possible that allows the function to be differentiable and continuous.