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Calculus/Differentiation/Basics of Differentiation/Exercises
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Some Important Theorems L'Hôpital's rule
Find the Derivative by Definition
Find the derivative of the following functions using the limit definition of the derivative.
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Prove the Constant Rule
10. Use the definition of the derivative to prove that for any fixed real number ,
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Find the Derivative by Rules
Find the derivative of the following functions:
Power Rule
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Product Rule
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Quotient Rule
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Chain Rule
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Exponentials
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Logarithms
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Trigonometric functions
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More Differentiation
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Implicit Differentiation
Use implicit differentiation to find y'
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Logarithmic Differentiation
Use logarithmic differentiation to find :
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Equation of Tangent Line
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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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).
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Higher Order Derivatives
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.
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Advanced Understanding of Derivatives
91. Let be the derivative of . Prove the derivative of is . 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 .
93. Let , where is the inverse of . Let be differentiable. What is ? Else, why can not be determined? .
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.
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Some Important Theorems L'Hôpital's rule