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DERIVATIVES UNIT PROBLEM SETS

PROBLEM SET #1 – Rate of Change ***Calculators Not Allowed***

Find the average rate of change for each function between the given values:

1. 𝑦 = 𝑥2 + 4𝑥 + 3 from 𝑥 = −2 to 𝑥 = 3

2. 𝑓(𝑥) = 𝑥3 − 𝑥2 + 1 from 𝑥 = −2 to 𝑥 = 2

3. 𝑔(𝑥) = 𝑠𝑖𝑛𝑥 from 𝑥 =𝜋

2 to 𝑥 = 𝜋

4. ℎ(𝑥) = 2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 from 𝑥 =𝜋

4 to 𝑥 =

3𝜋

4

5. 𝑟(𝑡) = √𝑡2 − 9 from 𝑡 = −5 and 𝑡 = 3

6. 𝑦 = log10 𝑡 from 𝑡 = 10 to 𝑡 = 100

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7. 𝑦 = 2 log5 𝑡 from 𝑡 = 1 to 𝑡 = 25

8. 𝑔(𝑡) = 𝑒𝑡 + 5 from 𝑡 = 0 and 𝑡 = 1

9. 𝑦 = |4 − 𝑥2| from 𝑥 = 0 to 𝑥 = 3

10. 𝑓(𝑥) =𝑥+2

𝑥−2 from 𝑥 = 3 to 𝑥 = 8

11. Andrew is a physics student testing the rate of change of objects he can throw. Given his calculations, if he throws the baseball from the top of a hill, it follows the equation 𝑥(𝑡) = −4.9𝑡2 + 14.7𝑡 + 25. He wants to know the average rate of change of the ball for each of the following time periods: (Calculator allowed) a) 𝑡 = 0 𝑡𝑜 𝑡 = 1.5 b) 𝑡 = 1.5 𝑡𝑜 𝑡 = 3 c) 𝑡 = 1 𝑡𝑜 𝑡 = 3

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PROBLEM SET #2 – Slope of a Curve ***Calculators Not Allowed***

For problems #1-8, find the limit of the function at the given point:

1. a) Find the derivative of the function 𝑦 = 4𝑥 + 1.

b) What is the value of the derivative at 𝑥 = 1 ?

2. a) Find 𝑓 ′(𝑥) if the function is 𝑓(𝑥) = 2𝑥2. b) What is the value at 𝑥 = 2 ?

3. a) Find the derivative of 𝑔(𝑥) = 3𝑥3 + 1.

b) What is the value of 𝑔′(1) ?

4. a) Find 𝑑𝑦

𝑑𝑥 of the function 𝑦 = (𝑥 + 1)2

b) What is the slope at 𝑥 = −1 ?

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5. a) Find ℎ′(𝑥) if ℎ(𝑥) = 2𝑥3 + 3𝑥2. b) What is the slope at 𝑥 = −2 ?

6. a) Find 𝑑𝑓

𝑑𝑟 if 𝑓(𝑟) = 𝜋𝑟2.

b) What is the slope at 𝑟 = 3 ?

7. a) Find the derivative of 𝑦 = √𝑥.

b) What is the slope at 𝑥 = 2 ?

8. Of the functions you have worked with, which type of functions have the same

average rate of change as their instantaneous rate of change? (i.e. Δ𝑦

Δ𝑥= 𝑦′) Try taking

the average rate of changes of the examples above. a) Linear b) Quadratic c) Cubic d) Square Root

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PROBLEM SET #3 – Derivative Rules ***Calculators Not Allowed***

1. Find the derivative of the function 𝑦 = 11.

2. Find 𝑓 ′(𝑥) if the function is 𝑓(𝑥) = 4𝑥2.

3. a) Find the derivative of 𝑔(𝑥) = 12𝑥3 − 4𝑥2 + 2𝑥.

b) What is the value of 𝑔′(−1) ?

4. Find 𝑑𝑦

𝑑𝑥 for the function 𝑦 = (𝑥 + 2)2.

5. a) Find ℎ′(𝑥) if ℎ(𝑥) = 2𝑥3/4.

b) What is the slope at 𝑥 = 16 ?

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6. a) Find 𝑑𝑓

𝑑𝑡 if 𝑓(𝑡) = 3√𝑡 .

b) What is the slope at 𝑡 = 2 ?

7. a) Find the derivative of 𝑦 = (2𝑥 + 1)(2𝑥 − 1) .

b) What is the slope at 𝑥 = 2 ?

8. a) Find 𝑔′(𝑥) if 𝑔(𝑥) = (𝑥2 + 4𝑥 + 1)2 .

b) What is the slope at 𝑥 = 0 ?

9. a) Find 𝑑𝑦

𝑑𝑥 if 𝑦 =

1

2√𝑥3

.

b) What is the slope at 𝑥 = 8 ?

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PROBLEM SET #4 – Higher Order Derivatives ***Calculators Not Allowed***

1. Find the 1st and 2nd derivative of the function 𝑦 = 20𝑥 .

2. Find 𝑓 ′(𝑥) and 𝑓 ′′(𝑥) of the function 𝑓(𝑥) = 13𝑥2 + 4𝑥 .

3. Find the 1st and 2nd derivative of 𝑔(𝑥) = (𝑥 + 4)3 .

4. Find 𝑑𝑦

𝑑𝑥 and

𝑑2𝑦

𝑑𝑥2 of the function 𝑦 = (𝑥 − 4)2 .

5. Find ℎ′′′(𝑥) if ℎ(𝑥) =1

24𝑥4 −

1

6𝑥3

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6. Find 𝑑2𝑓

𝑑𝑡2 if 𝑓(𝑡) = √𝑡 .

7. Find the derivative of 𝑦 = √𝑥3

.

8. Find the 1st and 2nd derivatives of 𝑓(𝑥) = 𝑥𝑏 a) where 𝑏 = 2 b) where 𝑏 = 3 c) generically for all 𝑏 > 3

9. Looking back at problem 8c above, we begin to see a pattern with derivatives. As we take derivatives, our exponents are used as coefficients and multiplied together. If we were to continue taking derivatives until the exponent is zero, we can see that the last coefficient is equal to the factorial (!) of the original power (any derivative after that will equal zero). If we stopped at any other time along that path, we would have a permutation of the exponents equal to 𝑎𝑃𝑑 where “a” is the original exponent and “d” is the specific derivative. Using this method, find the following derivatives: (calculator OK)

a) 8th derivative of 𝑥9 b) 10th derivative of 2𝑥10

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PROBLEM SET #5 – Trigonometry Rules ***Calculators Not Allowed***

1. a) Find the derivative of the function 𝑦 = 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 .

b) What is the value of the derivative at 𝑥 =𝜋

2 ?

c) What is the value of the derivative at 𝑥 =𝜋

6 ?

2. a) Find 𝑓 ′(𝑥) if the function is 𝑓(𝑥) =1

𝑠𝑖𝑛𝑥+

1

𝑐𝑜𝑠𝑥 .

b) What is the value of the derivative at 𝜋

4 ?

c) What is the value of the derivative at 𝜋

3?

3. a) Find the derivative of 𝑔(𝑥) = 3𝑡𝑎𝑛𝑥 − 4𝑐𝑜𝑡𝑥 .

b) What is the value of 𝑔′ (𝜋

6) ?

4. a) Find 𝑑𝑦

𝑑𝑥 of the function 𝑦 =

1

𝑐𝑜𝑡𝑥+

2

𝑐𝑜𝑡𝑥 .

b) What is the derivative at 𝑥 =𝜋

4 ?

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5. a) Find ℎ′(𝑥) if ℎ(𝑥) = −𝑐𝑠𝑐𝑥 + 𝑠𝑒𝑐𝑥 .

b) What is the slope at 𝑥 =5𝜋

6 ?

6. Find the first four derivatives of 𝑦 = 4𝑠𝑖𝑛𝑥 .

7. Find the first four derivatives of 𝑔(𝑥) = 5𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥

8. Find 𝑑𝑚

𝑑𝑥 if 𝑚(𝑥) = tan−1 𝑥 + cot−1 𝑥 .

9. a) Find the derivative of 𝑦 = 2 sin−1 𝑥 .

b) What is the value of the derivative at 𝑥 = 0.

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PROBLEM SET #6 – Product/Quotient Rule ***Calculators Not Allowed***

1. Find 𝑓 ′(𝑥) if the function is 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2) .

2. Find 𝑑𝑦

𝑑𝑥 of the function is 𝑦 = (𝑥2 + 4𝑥 − 3)(3𝑥2 − 10) .

3. Find the derivative of 𝑔(𝑥) = √𝑥(𝑥4 − 3𝑥2 − 10𝑥 + 1) .

4. Find ℎ′(𝑥) if ℎ(𝑥) = √𝑥3

𝑡𝑎𝑛𝑥 .

5. Find 𝑦′ if 𝑦 = 𝑐𝑜𝑠𝑥 ∙ 𝑠𝑖𝑛𝑥 .

6. Find 𝑓 ′(𝑥) if𝑓(𝑥) = sin2 𝑥 .

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7. Find the derivative of 𝑦 =𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥+

𝑐𝑜𝑠𝑥

𝑠𝑖𝑛𝑥 (without using trig shortcuts).

8. Find 𝑔′(𝑥) if 𝑔(𝑥) =𝑥2+1

𝑥2−1 .

9. Find 𝑑𝑦

𝑑𝑡 if 𝑦 =

𝑒𝑡+1

𝑡3 .

10. Using any prior rules, find the 1st and 2nd derivatives of 𝑦 = 𝑡𝑎𝑛𝑥 .

11. **Show using product rule that the derivative of sin3 𝑥 is 3 sin2 𝑥𝑐𝑜𝑠𝑥 .

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PROBLEM SET #7 Derivatives Using Tables ***Calculators Allowed***

x f(x) f’(x) g(x) g’(x)

-2 -19 16 -11 19

-1 -6 10 -2 4

0 1 4 -1 -1

1 2 -2 -2 6

2 -3 -8 1 7

1. Given ℎ(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥) find: ℎ′(2)

find: ℎ′(1)

2. Given 𝑗(𝑥) = 2𝑥2 ∙ 𝑓(𝑥) find: 𝑗′(0)

find: 𝑗′(2)

3. Given 𝑘(𝑥) =𝑓(𝑥)

𝑔(𝑥) find: 𝑘′(−1)

find: 𝑘′(1)

4. Given 𝑚(𝑥) = 𝑔(𝑥)2 find: 𝑚′(0)

find: 𝑚′(−2)

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t g(t) g’(t) h(t) h’(t)

0 -10 6 -1 -9

1 -3 1 -3 5

2 4 14 -1 -1/4

3 2 -1.5 -2 5

4 0 8 1/2 1

5. Given 𝑓(𝑡) = 𝑔(𝑡) ∙ ℎ(𝑡) find: 𝑓 ′(1)

find: 𝑓 ′(2)

6. Given 𝑟(𝑡) =𝑔(𝑡)

ℎ(𝑡) find: 𝑟 ′(0)

find: 𝑟 ′(3)

7. Given 𝑠(𝑡) = ℎ(𝑡)2 find: 𝑠′(0)

find: 𝑠′(4)

8. Given 𝑔(𝑡) = 2𝑡3ℎ(𝑡) find: 𝑔′(1)

find: 𝑔′(2)

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PROBLEM SET #8 – Tangent/Normal Lines ***Calculators Not Allowed***

For each question find the equations of the tangent & normal lines at the given value.

1. 𝑓(𝑥) = 4𝑥 + 7 at 𝑥 = 2

2. 𝑓(𝑥) = 3𝑥2 − 2𝑥 + 1 at 𝑥 = −3

3. 𝑦 = 3𝑐𝑜𝑠𝑥 + 1 at 𝑥 =𝜋

2

4. 𝑔(𝑥) = −4√𝑥 − 2 at 𝑥 = 4

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5. 𝑦 = 𝑐𝑜𝑠𝑥 ∙ 𝑠𝑖𝑛𝑥 at 𝑥 =𝜋

4

6. 𝑦 = |2𝑥| at 𝑥 = −3

7. ℎ(𝑥) =𝑥2+1

𝑥2−1 at 𝑥 = 0

8. 𝑓(𝑥) =1

𝑥 at 𝑥 = 2

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PROBLEM SET #9 – Derivatives of Log & e ***Calculators Not Allowed***

1. Find the derivative of: 𝑓(𝑥) = 5ln 𝑥 + 1

2. Find dy

dx for 𝑦 = 23𝑥 + 15𝑥 + 6𝑥

3. Differentiate: 𝑦 = 3 log4 𝑥 + 7 log2 𝑥

4. Find the derivative of: 𝑓(𝑥) = 𝑥2 + 2𝑥

5. Find dy

dx for: 𝑦 = 5𝑥 + 𝑒𝑥 + ln 𝑒

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6. Find 𝑦′ for: 𝑦 = ln 𝑥 + 𝑥2 log8 𝑥 + 𝑥3

7. Find the derivative of: 𝑓(𝑥) = 2𝑒𝑥 + 3𝑥4 ln 2 + 2𝑥

8. Find dy

dx for: 𝑦 =

3𝑥

log3 𝑥 +3𝑒𝑥

9. Differentiate: 𝑓(𝑥) = 5𝑥 + log7 𝑥 + 9ln 𝑥

10. Find the derivative of : 𝑓(𝑥) = 3𝑥 ln 𝑥 + 3𝑥2 ln 𝑥 + 6

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PROBLEM SET #10 – Chain Rule ***Calculators Not Allowed***

1. Find 𝑦′ for : 𝑦 = (5𝑥2 + 3𝑥)3

2. Given: 𝑓(𝑥) = 3𝑥(2𝑥 − 17)3 + 15 Find 𝑓′(𝑥)

3. Given: 𝑓(𝑥) = (4𝑥3 + 2𝑥2 + 1

2𝑥)5 Find 𝑓′(𝑥)

4. Find 𝑦′ for : 𝑦 = 3𝑐𝑜𝑠2𝑥

5. Differentiate: 𝑓(𝑥) = (sin (2𝑥3 − 1))2

6. Given: 𝑓(𝑥) = (3𝑥4 + 𝑥)(10𝑥2 − 𝑥)5 Find 𝑓′(𝑥)

7. Find 𝑑𝑦

𝑑𝑥 if : 𝑦 =

5𝑥2−3𝑥

(3𝑥7+2𝑥6)4

8. Differentiate: 𝑓(𝑥) = (15𝑥3 − 10𝑥5)3

2

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9. Given: 𝑓(𝑥) = (7𝑥 + (3𝑥 − 9)6)3 Find 𝑓′(𝑥)

10. Differentiate: 𝑓(𝑥) = 3tan (8𝑥)

11. Find 𝑑𝑦

𝑑𝑥 if 𝑦 = (−3𝑥3 + 2𝑥2 + 𝑥)4

12. Find 𝑦′ for : 𝑦 = 2𝑥8(4𝑥2 − 3𝑥 )2

13. Given: 𝑓(𝑥) = (7𝑥2 − 𝑥 + 15)−3 Find 𝑓′(𝑥)

14. Find 𝑑𝑦

𝑑𝑥 if 𝑦 = −2sin4(3𝑥 − 5)

15. Differentiate: 𝑓(𝑥) = (−4x2 − 6𝑥)2(𝑥5 + 5𝑥)3

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PROBLEM SET #11 – Derivs. of Inv. Functions ***Calculators Not Allowed***

1. If 𝑓(1) = 4 and 𝑓 ′(1) = 6 , find 𝑓−1′(4)

2. Find the derivative of the inverse of 𝑓(𝑥) = 𝑥4

3. If 𝑓(4) = 6 and 𝑓 ′(4) = 5 find 𝑓−1′(6)

4. If 𝑓(2) = 7 and 𝑓 ′(𝑥) = 3𝑥2 + 5𝑥 + 12 find 𝑓−1 ′(7)

5. If 𝑓(8) = 15 and 𝑓′(𝑥) = 𝑥3 + 2𝑥2 + 3𝑥 find 𝑓−1 ′(15)

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6. Find the derivative of the inverse of 𝑓(𝑥) = 𝑠𝑖𝑛𝑥

7. If 𝑓′(𝑥) =2𝑥+3

3𝑥+2 and 𝑓(3) = 6 find 𝑓−1 ′(6)

8. If 𝑓(2) = 23 and 𝑓′(𝑥) =5𝑥+8

5𝑥2+8 Find 𝑓−1 ′(23)

9. If 𝑓(𝑥) = 5𝑥 − 7 Find 𝑓−1 ′(8)

10. If 𝑓(1) = 5 and 𝑓(𝑥) = 2𝑥4 + 2𝑥3 + 2𝑥2 + 2𝑥 + 2 Find 𝑓−1 ′(5)

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PROBLEM SET #12 – Contin. vs. Differentiability ***Calculators Not Allowed***

1. If a function is differentiable on a given interval, it is also continuous.

a. True

b. False

2. For a function to be differentiable, it: (choose all that apply)

a. Must have no discontinuities

b. Can have discontinuities

c. Must have no vertical tangent lines

d. Can have vertical tangent lines

e. Must not have corners

f. May have cusps

3. At which values of x is 𝑓(𝑥) not differentiable?

4. At which values of x is 𝑓(𝑥) not differentiable?

5. At which values of x is 𝑓(𝑥) not differentiable?

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PROBLEM SET #13 – Der. of Piecewise & Abs. Value ***Calcs. Not Allowed***

1. a. What is the equivalent piecewise function for the following?

b. What is the derivative?

𝑦 = 2|𝑥 + 3| − 4

2. a. What is the equivalent piecewise function for the following?

b. What is the derivative?

𝑓(𝑥) = 4|2𝑥 − 2| + 5

3. Find the derivative of the following function:

𝑦 = {3(−𝑥 + 2) − 1 𝑥 ≤ 23(𝑥 − 2) − 1 𝑥 > 2

4. Find the derivative of the following function:

𝑦 = {5𝑥2 − 9𝑥 𝑥 < 1𝑙𝑛𝑥 𝑥 ≥ 1

5. What values of 𝑘 and 𝑚 will make the function differentiable over the interval (−1,10)?

𝑓(𝑥) = {𝑚𝑥 + 4 − 1 ≤ 𝑥 ≤ 2𝑘𝑥2 + 3 2 < 𝑥 ≤ 10

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6. What values of 𝑚 and 𝑗 will make the function differentiable over the interval (−3,9)?

𝑓(𝑥) = {𝑚𝑥2 + 5 − 3 ≤ 𝑥 ≤ 315 + 𝑗𝑥 3 < 𝑥 ≤ 9

7. What values of 𝑗 and 𝑘 will make the function differentiable over the interval (0,8)?

𝑓(𝑥) = {−𝑗 + 2𝑗𝑥 0 ≤ 𝑥 ≤ 12𝑘𝑥 + 15 1 < 𝑥 ≤ 8

For questions 8 – 10, choose all that apply:

8. 𝑓(𝑥) = |𝑥 + 23|

a. 𝑓(𝑥) is continuous at 𝑥 = −23

b. 𝑓(𝑥) is differentiable at 𝑥 = −23

c. 𝑓(𝑥) is not continuous at 𝑥 = −23

d. 𝑓(𝑥) is not differentiable at 𝑥 = −23

9. 𝑓(𝑥) = {4𝑥2 + 𝑥 𝑥 ≤ 12𝑥4 + 𝑥 𝑥 > 1

a. 𝑓(𝑥) is continuous at 𝑥 = 1

b. 𝑓(𝑥) is differentiable at 𝑥 = 1

c. 𝑓(𝑥) is not continuous at 𝑥 = 1

d. 𝑓(𝑥) is not differentiable at 𝑥 = 1

10. 𝑓(𝑥) = {𝑥2 − 4𝑥 + 8 𝑥 ≤ 32𝑥 − 1 𝑥 > 3

a. 𝑓(𝑥) is continuous at 𝑥 = 3

b. 𝑓(𝑥) is differentiable at 𝑥 = 3

c. 𝑓(𝑥) is not continuous at 𝑥 = 3

d. 𝑓(𝑥) is not differentiable at 𝑥 = 3

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PROBLEM SET #14 – Implicit Differentiation ***Calculators Not Allowed***

1. Find 𝑑𝑦

𝑑𝑥: 𝑦2+ 𝑥2 = 2𝑥 − 5𝑦3

2. Find 𝑑𝑦

𝑑𝑡: 𝑦 = 3𝑡 + 2𝑡𝑦 + 𝑡2

3. Find 𝑑𝑥

𝑑𝑡: 15𝑡 = 2𝑥2 + 4𝑥

4. Find 𝑑𝑦

𝑑𝑥:

3𝑥

2𝑦+ 2𝑥2 + 4𝑦 = 5𝑥

5. Find 𝑑𝑉

𝑑𝑡: 𝑉 =

4

3𝜋𝑟3

6. Find 𝑑𝑦

𝑑𝑥: 𝑥4 + 𝑥𝑦4 = 4𝑥

7. Find 𝑑𝑧

𝑑𝑦: 3𝑦3 + 3𝑧2 = 5𝑧

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8. Find 𝑑𝑦

𝑑𝑥: sin 𝑥 = 2𝑥 cos 𝑦 + 2

9. Find 𝑑𝜃

𝑑𝑡: 2 sin2 𝜃 + cos2 𝜃 = 2𝑡

10. Find 𝑑𝐴

𝑑𝑡: 𝐴 = 2𝜋𝑟ℎ + 2𝜋𝑟2

11. Find 𝑑𝑦

𝑑𝑥: sin 2𝑥 + cos 𝑦 = 5

12. Find the slope of the tangent line at the point (1,2) for: 3𝑦2 + 5𝑥3 = 17

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13. Find the slope of the tangent line at the point (16,-1) for: 2𝑦2 + √𝑥 − 6 = 0

14. Find the slope of the tangent line at the point (1,1) for: 2𝑥3 = 3𝑥2 − 3𝑥𝑦 + 2

15. Find the equation of the tangent line at the point (4,8) for: 4x2 − 2𝑦2 = −16𝑥

16. Find the equation of the tangent line at the point (1,0) for: 1

𝑥+ 5 = 𝑦𝑥 + 6

17. Find the equation of the tangent line at the point (-3,3) for: 𝑥𝑦2 + 𝑦 = 8𝑥

NJCTL.org

Limits and Continuity- Answer Keys Problem Set #1 – Rate of Change

1. 5 2. 4

3. −2

𝜋

4. −4

𝜋

5. −1

2

6. 1

90

7. 1

6

8. 𝑒 − 1

9. 1

3

10. −2

3

11. a. 7.35 b. −7.35 c. −4.9

Problem Set #2 – Slope of a Curve

1. a. 𝑦′ = 4 b. 4

2. a. 𝑓′(𝑥) = 4𝑥 b. 𝑓′(2) = 8

3. a. 𝑔′(𝑥) = 9𝑥2 b. 𝑔′(1) = 9

4. a. 𝑑𝑦

𝑑𝑥= 2𝑥 + 2

b. 0 5. a. ℎ′(𝑥) = 6𝑥2 + 6𝑥

b. ℎ′(2) = 12

6. a. 𝑑𝑓

𝑑𝑟= 2𝜋𝑟

b. 6𝜋

7. a. 𝑦′ =1

2√𝑥

b. √2

4

8. a. linear

Problem Set #3 – Derivative Rules

1. 𝑦′ = 0 2. 𝑓′(𝑥) = 8𝑥 3. a. 𝑔′(𝑥) = 36𝑥2 − 8𝑥 + 2

b. 𝑔′(−1) = 46

4. 𝑑𝑦

𝑑𝑥= 2𝑥 + 4

5. a. 3

2 √𝑥4

b. 3

4

6. a. 𝑑𝑓

𝑑𝑡=

3

2√𝑡

b. 3√2

4

7. a. 𝑦′ = 8𝑥 b. 16

8. a. 𝑔′(𝑥) = 4𝑥3 + 24𝑥2 +36𝑥 + 8 b. 𝑔′(0) = 8

9. a. 𝑑𝑦

𝑑𝑥=

1

6 √𝑥23

b. 1

24

Problem Set #4 – Higher Order Derivatives

1. 𝑦′ = 20 𝑦′′ = 0 2. 𝑓′(𝑥) = 26𝑥 + 4

𝑓′′(𝑥) = 26 3. 𝑔′(𝑥) = 3𝑥2 + 24𝑥 + 48

𝑔′′(𝑥) = 6𝑥 + 24

4. 𝑑𝑦

𝑑𝑥= 2𝑥 − 8

𝑑2𝑦

𝑑𝑥2= 2

5. ℎ′′′(𝑥) = 𝑥 − 1

6. 𝑑2𝑓

𝑑𝑡2=

−1

4√𝑡3

NJCTL.org

7. 𝑦′ =1

3 √𝑥23

8. a. 𝑓′(𝑥) = 2𝑥 𝑓′′(𝑥) = 2 b. 𝑓′(𝑥) = 3𝑥2 𝑓′′(𝑥) = 6𝑥 c. 𝑓′(𝑥) = 𝑏𝑥𝑥−1

𝑓′′(𝑥) = (𝑏 − 1)𝑏𝑥𝑏−2 9. a. 362,880𝑥

b. 7,257,600

Problem Set #5 – Trigonometry Rules

1. a. 𝑦′ = 𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥

b. 𝑦′ (𝜋

2) = −1

𝑦′ (𝜋

6) =

√3−1

2

2. a. 𝑓′(𝑥) = −𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥 +𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥

b. 𝑓′ (𝜋

4) = 0

𝑓′ (𝜋

3) =

−2

3+ 2√3

3. a. 𝑔′(𝑥) = 3 sec2 𝑥 + 4 csc2 𝑥

b. 𝑔′ (𝜋

6) = 20

4. a. 𝑑𝑦

𝑑𝑥= 3 sec2 𝑥

b. 6 5. a. ℎ′(𝑥) = 𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥 +

𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥

b. −2√3 +2

3

6. 𝑦′ = 4𝑐𝑜𝑠𝑥 𝑦′′ = −4𝑠𝑖𝑛𝑥 𝑦′′′ = −4𝑐𝑜𝑠𝑥 𝑦4 = 4𝑠𝑖𝑛𝑥

7. 𝑔′(𝑥) = −5𝑠𝑖𝑛𝑥 − 𝑐𝑜𝑠𝑥 𝑔′′(𝑥) = −5𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 𝑔′′′(𝑥) = 5𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 𝑔4(𝑥) = 5𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥

8. 𝑑𝑚

𝑑𝑥= 0

9. a. 2

√1−𝑥2

b. 2

Problem Set #6– Product/Quotient Rule

1. 𝑓′(𝑥) = 2𝑥

2. 𝑑𝑦

𝑑𝑥= 12𝑥3 + 36𝑥2 − 38𝑥 −

40

3. 𝑔′(𝑥) =9

2𝑥7/2 −

15

2𝑥3/2 −

15𝑥1/2 +1

2𝑥−1/2

4. ℎ′(𝑥) =1

3𝑥−2/3𝑡𝑎𝑛𝑥 +

𝑥1/3 sec2 𝑥 5. 𝑦′ = − sin2 𝑥 + cos2 𝑥 or 𝑦′ =

𝑐𝑜𝑠2𝑥

6. 𝑑𝑓

𝑑𝑥= 2𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑥 or 𝑦′ = 𝑠𝑖𝑛2𝑥

7. 𝑦′ = sec2 𝑥 + csc2 𝑥

8. 𝑔′(𝑥) =−4𝑥

𝑥4−2𝑥2+1

9. 𝑑𝑦

𝑑𝑡=

𝑡𝑒𝑡−3𝑒𝑡−3

𝑡4

10. sec2 𝑥 2 sec2 𝑥𝑡𝑎𝑛𝑥

11. must show work

Problem Set #7 – Derivative Tables

1. ℎ′(2) = −29 ℎ′(1) = 16

2. 𝑗′(0) = 0 𝑗′(2) = −88

3. 𝑘′(−1) = 1 𝑘′(1) = −2

4. 𝑚′(0) = 2 𝑚′(−2) = −418

5. 𝑓′(1) = −18 𝑓′(2) = −15

6. 𝑟′(0) = −96

𝑟′(3) = −7

4

7. 𝑠′(0) = 18 𝑠′(4) = 1

8. 𝑔′(1) = −8 𝑔′(2) = −28

NJCTL.org

Problem Set #8 – Tangent/Normal Lines

1. tangent: 𝑦 − 15 = 4(𝑥 − 2) or 𝑦 = 4𝑥 + 7

normal: =𝑦 − 15 = −1

4(𝑥 − 2)

or 𝑦 = −1

4𝑥 +

31

2

2. tangent: 𝑦 − 34 = −20(𝑥 + 3) or 𝑦 = −20𝑥 − 26

normal: 𝑦 − 34 =1

20(𝑥 + 3) or

𝑦 =1

20𝑥 +

683

20

3. tangent: 𝑦 − 1 = −3 (𝑥 −𝜋

2)

or 𝑦 = −3𝑥 +3𝜋+2

2

normal: 𝑦 − 1 =1

3(𝑥 −

𝜋

2) or

𝑦 =1

3𝑥 +

6−𝜋

6

4. tangent: 𝑦 + 10 = −(𝑥 − 4) or 𝑦 = −𝑥 − 6 normal 𝑦 + 10 = 1(𝑥 − 4) or 𝑦 = 𝑥 − 14

5. tangent: 𝑦 =1

2

normal: 𝑥 =𝜋

4

6. at 𝑥 = −3: tangent: 𝑦 = −2𝑥

normal: 𝑦 − 6 =1

2(𝑥 + 3) or

𝑦 =1

2𝑥 +

15

2

at 𝑥 = 3: tangent: 𝑦 = 2𝑥

normal: 𝑦 − 6 = −1

2(𝑥 − 3) or

𝑦 = −1

2𝑥 +

15

2

7. tangent: 𝑦 = −1 normal: 𝑥 = 0

8. tangent: 𝑦 −1

2= −

1

4(𝑥 − 2) or

𝑦 = −1

4𝑥 + 1

normal: 𝑦 −1

2= 4(𝑥 − 2) or

𝑦 = 4𝑥 −15

2

Problem Set #9 – Derivatives of Logs and e

1. 𝑓′(𝑥) =5

𝑥

2. 𝑑𝑦

𝑑𝑥= 23𝑥𝑙𝑛23 + 15𝑥𝑙𝑛15 +

6𝑥𝑙𝑛6

3. 𝑦′ =3

𝑥𝑙𝑛4+

7

𝑥𝑙𝑛2

4. 𝑓′(𝑥) = 2𝑥 + 2𝑥𝑙𝑛2

5. 𝑑𝑦

𝑑𝑥= 5𝑥𝑙𝑛5 + 𝑒𝑥

6. 𝑦′ = 3𝑥2 + 2𝑥 log8 𝑥 +𝑥

𝑙𝑛8+

1

𝑥

7. 𝑓′(𝑥) = 2𝑒𝑥 + 24𝑥3𝑙𝑛2 +2𝑥𝑙𝑛2

8. 𝑑𝑦

𝑑𝑥=

(3𝑥𝑙𝑛3)(log3 𝑥+3𝑒𝑥)−(3𝑥)(1

𝑥𝑙𝑛3+3𝑒𝑥)

(log3 𝑥+3𝑒𝑥)2

9. 𝑓′(𝑥) = 5𝑥𝑙𝑛5 +1

𝑥𝑙𝑛7+

9

𝑥

10. 𝑓′(𝑥) = 3𝑥𝑙𝑛3𝑙𝑛𝑥 +3𝑥

𝑥+

6𝑥𝑙𝑛𝑥 + 3𝑥 Problem Set #10 – Chain Rule

1. 3(5x2 + 3𝑥)2(10𝑥 + 3)

2. 3(2𝑥 − 17)3 + 18𝑥(2𝑥 − 17)2

3. 5 (4𝑥3 + 2𝑥2 +1

2𝑥)

4

(12𝑥2 + 4𝑥 +1

2)

4. −6𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥

5. 12𝑥2 sin(2𝑥3 − 1) cos (2𝑥3 − 1)

6. (12𝑥3 + 1)(10𝑥2 − 𝑥)5 + 5(3𝑥4 +

𝑥)(10𝑥2 − 𝑥)4(20𝑥 − 1)

7. (10𝑥−3)(3𝑥7+2𝑥6)4−4(5𝑥2−3𝑥)(3𝑥7+2𝑥6)3(21𝑥6+12𝑥5)

(3𝑥7+2𝑥6)8

8. 3

2(15𝑥3 − 10𝑥5)

1

2 (45𝑥2 − 50𝑥4) or 3

2(45x2 − 50𝑥4)√15𝑥3 − 10𝑥5

NJCTL.org

9. 9(7𝑥 + (3𝑥 − 9)6)2 (7 + 6(3𝑥 −

9)5)

10. 24 𝑠𝑒𝑐2 8𝑥

11. 4(−3𝑥3 + 2𝑥2 + 𝑥)3 (−9𝑥2 +

4𝑥 + 1)

12. 16𝑥7(4𝑥2 − 3𝑥)2 + 4𝑥8(4𝑥2 −

3𝑥)(8𝑥 − 3)

13. −42𝑥+3

(7𝑥2−𝑥+15)4

14. −24𝑠𝑖𝑛3(3𝑥 − 5) cos (3𝑥 − 5)

15. 2(−4x2 − 6𝑥)(−8𝑥 − 6)(𝑥5 +

5𝑥)3 + 3(−4𝑥2 − 6𝑥)2(𝑥5 +

5𝑥)2(5𝑥4 + 5)

Problem Set #11 – Der. of Inverse Fns.

1. 1

6

2. 1

4 √𝑥34 or 1

4𝑥−

3

4

3. 1

5

4. 1

34

5. 1

664

6. 1

√1−𝑥2

7. 11

9

8. 14

9

9. 1

5

10. 1

20

Problem Set #12 – Continuity vs. Diff.

1. True

2. a, c, e

3. 𝑥 = −4, 𝑥 = −1

4. 𝑥 = −1, 𝑥 = 4, 𝑥 = 6

5. 𝑥 = −4, 𝑥 = −3, 𝑥 = −2, 𝑥 = 3

Problem Set #13 – Derivatives of Piecewise & Absolute Value Functions 1.

a. 𝑦 = {−2𝑥 − 10 𝑥 ≤ −32𝑥 + 2 𝑥 > −3

b. 𝑦′ = {−2 𝑥 < −32 𝑥 > −3

2.

a. 𝑦 = {−8𝑥 + 13 𝑥 ≤ 18𝑥 − 3 𝑥 > 1

b. 𝑦′ = {−8 𝑥 < 18 𝑥 > 1

3. 𝑦′ = {−3 𝑥 < 23 𝑥 > 2

4. 𝑦′ = {10𝑥 − 9 𝑥 ≤ 11

𝑥 𝑥 > 1

5. 𝑘 = −1

4 , 𝑚 = −1

6. 𝑗 = −20

3, 𝑚 = −

10

9

7. 𝑗 = −15, 𝑘 = −15

8. a, d

9. c, d

10. a, b

Problem Set #14 – Implicit Differentiation

1. dy

dx=

2−2𝑥

2𝑦+15𝑦2

2. dy

dt=

3+2𝑦+2𝑡

1−2𝑡

3. 𝑑𝑥

𝑑𝑡=

15

4𝑥+4

4. dy

dx=

10𝑦2−8𝑥𝑦2−3𝑦

8𝑦2−3𝑥

5. 𝑑𝑉

𝑑𝑡= 4𝜋𝑟2 𝑑𝑟

𝑑𝑡

6. dy

dx=

4𝑥3+𝑦4−4

−4𝑥𝑦3

7. 𝑑𝑧

𝑑𝑦=

9𝑦2

5−6𝑧

8. dy

dx=

cos 𝑥−2 cos 𝑦

−2𝑥 sin 𝑦

NJCTL.org

9. 𝑑𝜃

𝑑𝑡=

1

sin 𝜃 cos 𝜃

10. 𝑑𝐴

𝑑𝑡= 2𝜋ℎ

𝑑𝑟

𝑑𝑡+ 2𝜋𝑟

𝑑ℎ

𝑑𝑡+ 4𝜋𝑟

𝑑𝑟

𝑑𝑡

11. dy

dx=

2 cos 2𝑥

sin 𝑦

12. 𝑚 = −5

4

13. 𝑚 =1

32

14. 𝑚 = −1

15. 𝑦 − 8 =3

2(𝑥 − 4) or 𝑦 =

3

2𝑥 + 2

16. 𝑦 = −𝑥 + 1

17. 𝑦 − 3 =1

17(𝑥 + 3) 𝑜𝑟 𝑦 =

1

17𝑥 +

54

17