Calculus/Differentiation/Basics of Differentiation/Solutions

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Find the Derivative by Definition[edit]

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Prove the Constant Rule[edit]

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

Find the Derivative by Rules[edit]

Power Rule[edit]

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Product Rule[edit]

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Quotient Rule[edit]

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Chain Rule[edit]

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Let . Then

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Let . Then

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Let . Then

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Let . Then



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Let . Then



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Let . Then



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Let . Then



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Let . Then



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Let . Then



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Let . Then



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Let . Then

Exponentials[edit]

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Let . Then

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Let

Then

Using the chain rule, we have

The individual factor are

So

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Logarithms[edit]

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Let . Then

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Trigonometric functions[edit]

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More Differentiation[edit]

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Let . Then


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Implicit Differentiation[edit]

Use implicit differentiation to find y'

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Logarithmic Differentiation[edit]

Use logarithmic differentiation to find :

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Equation of Tangent Line[edit]

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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a)
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a)
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a)
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a)
b)


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a)
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a)
/ b)


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






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






Higher Order Derivatives[edit]

77. What is the second derivative of ?



78. 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.