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Calculus 1 Tutor
Worksheet 14
Derivatives
and Integrals
of Logarithms
2
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Calculus 1 Tutor - Worksheet 14 – Derivatives of Logarithms
1. Suppose
. Find .
2. If
, what is ?
3. Find if
.
3
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4. If
, what is ?
5. Suppose
. Find .
6. What is if
?
4
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7. Suppose
. What is ?
8. Find if
?
9. What is if
?
5
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10. Suppose
, what is ?
11. Evaluate
12. Evaluate
6
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13. Evaluate
.
14. Evaluate
.
15. Evaluate the definite integral.
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16. Evaluate
.
17. Evaluate
.
18. Evaluate the definite integral.
8
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19. Evaluate the definite Integral.
20. Evaluate the definite integral.
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Answers - Calculus 1 Tutor - Worksheet 14 – Derivatives of Logarithms
1. Suppose
. Find .
The formula for the derivative of the natural logarithm function is:
and
Therefore,
Answer:
2. If
, what is ?
The formula for the derivative of the natural logarithm function is:
and
Therefore,
Simplify:
Answer:
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3. Find if
. The formula for the derivative of the natural
logarithm function is:
and
Therefore,
Answer:
4. If
, what is ? The formula for the derivative of the
natural logarithm function is:
and
Therefore,
Answer:
5. Suppose
. Find . The formula for the derivative of the
natural logarithm function is:
and
Therefore, using this rule and the product rule,
Answer:
11
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6. What is if
? The formula for the derivative of the
natural logarithm function is:
and
Therefore, using this rule and the product rule,
Simplify:
Answer:
7. Suppose
. What is ? The formula for the derivative of
the natural logarithm function is:
and
Therefore, using this rule and the quotient rule,
This expression does not simplify.
Answer:
12
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8. Find if
? The formula for the derivative
of the natural logarithm function is:
and
Therefore, using this rule and the product rule,
Simplify:
Answer:
9. What is if
? The formula for the derivative of
the natural logarithm function is:
and
Therefore, using this rule and the product rule,
Answer:
13
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10. Suppose
, what is ? The formula for the
derivative of the natural logarithm function is:
and
Therefore, using this rule and the product rule,
Answer:
11. Evaluate
Integration involving the natural logarithmic function follows the pattern:
and
Therefore, let and . Thus,
Answer:
14
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12. Evaluate
and
Therefore, let
and . Thus,
Answer:
13. Evaluate
.
and
Therefore, let and . Thus, Move the constant 5 out of the
integral and multiply by
.
Answer:
15
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14. Evaluate
.
and
Therefore, let and
.
Answer:
15. Evaluate the definite integral.
and
Let
and . Then, since the integral is a definite integral,
change the integration limits so they are in terms of :
and
. Therefore,
Answer:
16
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16. Evaluate
.
and
Since
, let and . Therefore,
Answer:
17. Evaluate
.
Change the integrand from to
. Then, let and .
Now, integrate using the rule:
and
The integral needs a negative inside so multiply the integral by a double negative:
Answer:
17
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18. Evaluate the definite integral.
and
Therefore, let
and . Then, since the integral is a
definite integral, change the integration limits so they are in terms of .
and
Therefore,
Answer:
18
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19. Evaluate the definite Integral.
and
Since
, let and . Then since the integral is a
definite integral, change the integration limits so they are in terms of .
and
Therefore,
Answer:
19
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20. Evaluate the definite integral.
and
Let
and . Then, because the integral is a
definite integral, change the integration limits so they are in terms of .
and
. Therefore,
Answer:
