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Calculus - Derivatives - Review - Multiple choice test 4

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Question 1. Define[eq1]The derivative of $fleft( xight) $ is:

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Explanation. The function $fleft( xight) $ is a composite function:[eq6]where:[eq7]and[eq8]We need to use the chain rule for the derivative of a composite function:[eq9]The derivative of $hleft( xight) $ is:[eq10]The derivative of $gleft( yight) $ is:[eq11]which evaluated at [eq12] gives:[eq13]Therefore:[eq14]See the calculus review page entitled Derivatives - Review for more details.

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Question 2. Define[eq15]The derivative of $fleft( xight) $ is:

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Explanation. We need to use the rule for differentiating a product of functions:[eq20]See the calculus review page entitled Derivatives - Review for more details.

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Question 3. Define[eq21]The derivative of $fleft( xight) $ is:

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Explanation. The function $fleft( xight) $ is a composite function:[eq26]where:[eq27]and[eq28]We need to use the chain rule for the derivative of a composite function:[eq29]The derivative of $hleft( xight) $ is:[eq30]The derivative of $gleft( yight) $ is:[eq31]which evaluated at [eq32] gives:[eq33]Therefore:[eq34]See the calculus review page entitled Derivatives - Review for more details.

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Question 4. Define[eq35]The derivative of $fleft( xight) $ is:

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Explanation. The function $fleft( xight) $ is a trigonometric function multiplied by a constant ($pi $):[eq40]See the calculus review page entitled Derivatives - Review for more details.

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Question 5. Define[eq41]The derivative of $fleft( xight) $ is:

nav_button [eq42]

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Explanation. We need to use the rule for differentiating a product of functions:[eq46]See the calculus review page entitled Derivatives - Review for more details.

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Question 6. Define[eq47]The derivative of $fleft( xight) $ is:

nav_button [eq48]

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nav_button Explanation | nav_button Next question

Explanation. The function $fleft( xight) $ is a composite function:[eq52]where:[eq53]and[eq54]We need to use the chain rule for the derivative of a composite function:[eq55]The derivative of $hleft( xight) $ is:[eq56]The derivative of $gleft( yight) $ is:[eq57]which evaluated at [eq58] gives:[eq59]Therefore:[eq60]See the calculus review page entitled Derivatives - Review for more details.

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Question 7. The function[eq61]is the derivative of:

nav_button [eq62]

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Explanation. The function $fleft( xight) $ is a trigonometric function ([eq66]) multiplied by a constant ($-1$). Its derivative is:[eq67]See the calculus review page entitled Derivatives - Review for more details.

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Question 8. The function:[eq68]is the derivative of:

nav_button [eq69]

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Explanation. The function $fleft( xight) $ is a composite function:[eq73]where:[eq74]and[eq75]We need to use the chain rule for the derivative of a composite function:[eq76]The derivative of $hleft( xight) $ is:[eq77]The derivative of $gleft( yight) $ is:[eq78]which evaluated at [eq79] gives:[eq80]Therefore:[eq81]See the calculus review page entitled Derivatives - Review for more details.

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Question 9. The function:[eq82]is the derivative of:

nav_button [eq83]

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Explanation. We need to use the rule for differentiating a linear combination of functions:[eq87]See the calculus review page entitled Derivatives - Review for more details.

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Question 10. The function:[eq88]is the derivative of:

nav_button [eq89]

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nav_button [eq91]

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nav_button Explanation

Explanation. The function $fleft( xight) $ is a composite function:[eq93]where:[eq94]and[eq95]We need to use the chain rule for the derivative of a composite function:[eq96]The derivative of $hleft( xight) $ is:[eq97]The derivative of $gleft( yight) $ is:[eq98]which evaluated at [eq99] gives:[eq100]Therefore:[eq101]See the calculus review page entitled Derivatives - Review for more details.

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