Calculus/Differentiation/Basics of Differentiation/Exercises

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Differentiation/Basics of Differentiation/Exercises

Find the Derivative by Definition[edit | edit source]

Find the derivative of the following functions using the limit definition of the derivative.

1.
2.
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9.

Solutions

Prove the Constant Rule[edit | edit source]

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

Solutions

Find the Derivative by Rules[edit | edit source]

Find the derivative of the following functions:

Power Rule[edit | edit source]

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Solutions

Product Rule[edit | edit source]

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Solutions

Quotient Rule[edit | edit source]

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Solutions

Chain Rule[edit | edit source]

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Solutions

Exponentials[edit | edit source]

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Solutions

Logarithms[edit | edit source]

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Solutions

Trigonometric functions[edit | edit source]

63.
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Solutions

More Differentiation[edit | edit source]

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Solutions

Implicit Differentiation[edit | edit source]

Use implicit differentiation to find y'

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Solutions

Logarithmic Differentiation[edit | edit source]

Use logarithmic differentiation to find :

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Solutions

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.

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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).

Solutions

Higher Order Derivatives[edit | edit source]

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.

Solutions

Advanced Understanding of Derivatives[edit | edit source]

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?
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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) .
(b) There is no that allows the following to be true. Proof in solutions.
(a) .
(b) There is no that allows the following to be true. Proof in solutions.

Solutions

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Differentiation/Basics of Differentiation/Exercises