TBIL-Calc Differentiation strategy (DF6)

Imagine that you are sorting functions into different categories based on how you would differentiate them. In what category (or categories) might these functions fall?

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Remark 2.6.2.

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To take a derivative, we need to examine how the function is built and then proceed accordingly. Below are some questions you might ask yourself as you take the derivative of a function, especially one where multiple rules might need to be used:

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How is this function built algebraically? What kind of function is this? What is the big picture?
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Where do you start?
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Is there an easier or more convenient way to write the function?
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Are there products or quotients involved?
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Is this function a composition of two (or more) elementary functions? If so, what are the outside and inside functions?
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What derivative rules will be needed along the way?

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Activity 2.6.3.

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Consider the function

f (x) = x^{3} \sqrt{3 - 8 x^{2}} .

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(a)

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You will need multiple derivative rules to find

f^{'} (x) .

Which rule would need to be applied first? In other words, what is the big picture here?

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Chain rule
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Power rule
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Product rule
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Quotient rule
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Sum/difference rule

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(b)

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What other rules would be needed along the way? Select all that apply.

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Chain rule
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Power rule
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Product rule
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Quotient rule
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Sum/difference rule

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(c)

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Write an outline of the steps needed if you were asked to take the derivative of

f (x) .

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Activity 2.6.4.

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Consider the function

f (x) = {(\frac{\ln x}{(3 x - 4)^{3}})}^{5} .

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(a)

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You will need multiple derivative rules to find

f^{'} (x) .

Which rule would need to be applied first? In other words, what is the big picture here?

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Chain rule
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Power rule
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Product rule
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Quotient rule
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Sum/difference rule

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(b)

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What other rules would be needed along the way? Select all that apply.

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Chain rule
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Power rule
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Product rule
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Quotient rule
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Sum/difference rule

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(c)

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Write an outline of the steps needed if you were asked to take the derivative of

f (x) .

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Activity 2.6.5.

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Consider the function

f (x) = \sin (\cos (\tan (2 x^{3} - 1))) .

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(a)

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You will need multiple derivative rules to find

f^{'} (x) .

Which rule would need to be applied first? In other words, what is the big picture here?

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Chain rule
🔗
Power rule
🔗
Product rule
🔗
Quotient rule
🔗
Sum/difference rule

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(b)

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What other rules would be needed along the way? Select all that apply.

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Chain rule
🔗
Power rule
🔗
Product rule
🔗
Quotient rule
🔗
Sum/difference rule

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(c)

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Write an outline of the steps needed if you were asked to take the derivative of

f (x) .

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Activity 2.6.6.

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Consider the function

f (x) = \frac{x^{2} e^{x}}{2 x^{3} - 5 x + \sqrt{x}} .

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(a)

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You will need multiple derivative rules to find

f^{'} (x) .

Which rule would need to be applied first? In other words, what is the big picture here?

🔗
Chain rule
🔗
Power rule
🔗
Product rule
🔗
Quotient rule
🔗
Sum/difference rule

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(b)

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What other rules would be needed along the way? Select all that apply.

🔗
Chain rule
🔗
Power rule
🔗
Product rule
🔗
Quotient rule
🔗
Sum/difference rule

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(c)

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Write an outline of the steps needed if you were asked to take the derivative of

f (x) .

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Activity 2.6.7.

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Find the derivative of the following functions. For each, include an explanation of the steps involved that references the algebraic structure of the function.

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(a)

f (x) = e^{5 x} (x^{2} + 7^{x})^{3}

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(b)

f (x) = {(\frac{3 x + 1}{2 x^{6} - 1})}^{5}

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(c)

f (x) = \sqrt{\cos (2 x^{2} + x)}

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(d)

f (x) = \tan (x e^{x})

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Activity 2.6.8.

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Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (constant multiple, sum/difference, etc.) you are using in your work.

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