Welcome!

Today’s Topic is: Derivatives

DerivativesLimitDefinition & Equation of the Tangent Line

Basic Derv. (including trig)

Product and Quotient Rule (including trig)

Chain Rule (including trig)

Complex Derv. (including higher order)

200 200 200 200 200

400 400 400 400 400

600 600 600 600 600

800 800 800 800 800

1000 1000 1000 1000 1000

Bonus Question: 5000 pts

Limit Defn & y=mx+b: 200• Question: Determine the correct function to find the

derivative using the limit definition. 1. h(𝑥) = 4𝑥4 + 5𝑥 − 7

a. ℎ′(𝑥) =4 𝑥+ℎ 4+5 𝑥+ℎ −7− 4𝑥4+5𝑥−7

ℎ

b. ℎ′(𝑥) = limℎ→0

4 𝑥+ℎ 4+5 𝑥+ℎ −7− 4𝑥4+5𝑥−7

ℎ

c. ℎ′(𝑥) = limℎ→0

4𝑥4+5𝑥−7+ℎ− 4𝑥4+5𝑥−7

ℎ

d. ℎ′(𝑥) = limℎ→0

4 𝑥+ℎ 4+5 𝑥+ℎ −7−4𝑥4+5𝑥−7

ℎ

• Answer: • B

Topic 1: 400

• Question:

• Determine the correct set-up of the derivative

using the limit definition for: 𝑔(𝑥) =𝑥

𝑥−2

• Answer

• 𝑔′(𝑥) = limℎ→0

𝑥+ℎ

𝑥+ℎ−2−

𝑥

𝑥−2

ℎ

Topic 1: 600

• Question:

• Determine the slope of the tangent line for

𝑓(𝑥) = 4 − 𝑥2 𝑎𝑡 𝑥 = −1.

• Answer

• 2

Topic 1: 800

• Question:

• Write the equation of the tangent line for 𝑓(𝑥) = Τ1 2 𝑥

2 − 3𝑥 − 2 𝑎𝑡 x = 1 .

• Answer

• y = -2x – 2.5

Topic 1: 1000

• Question:

• Find the equation of the tangent line for

𝑓 𝑥 = sin 3𝑥 − 5 2 at 𝑥 =5𝜋

6.

• Answer

• y = -4.917x + 13.831

Topic 2: 200

• Question:

• Determine the derivative of

𝑦 = 5𝑥−4 −7

8𝑥−2 + 3𝑥2 − 6.

• Answer5 37

20 64

y x x x

Topic 2: 400

• Question:

• Determine the derivative of

𝑦 =3

4𝑥3+

7

2𝑥9+

52𝑥4 −

83𝑥9

• Answer

•𝑑𝑦

𝑑𝑥= −

9

4𝑥−4 −

63

2𝑥−10 +

4

5

52𝑥 Τ−1

5 −9

8

83𝑥 Τ1 8

Topic 2: 600• Question:• Determine the following:

𝑑𝑥 sin𝑥 = 𝑑𝑥 csc𝑥 =

𝑑𝑥 cos𝑥 = 𝑑𝑥 sec𝑥 =

𝑑𝑥 tan𝑥 = 𝑑𝑥 cot𝑥 =• Answer

𝑑𝑥 sin𝑥 = cos𝑥 𝑑𝑥 csc𝑥 = −csc𝑥cot𝑥

𝑑𝑥 cos𝑥 = −sin𝑥 𝑑𝑥 sec𝑥 = sec𝑥tan𝑥

𝑑𝑥 tan𝑥 = sec2𝑥 𝑑𝑥 cot𝑥 = −csc2𝑥

Topic 2: 800

• Question:

• Determine the derivative of 𝑓 𝑥 = 4𝜋csc𝑥

• Answer

• 𝑓′ 𝑥 = −4𝜋csc𝑥cot𝑥

Topic 2: 1000

• Question:

• Determine the derivative of the following in the simplest form. 𝑦 = sin𝑥sec𝑥

• Answer

•𝑑𝑦

𝑑𝑥= sec2𝑥

Topic 3: 200

• Question:

• Determine the derivative of 𝑔 𝑥 = 𝑥2 + 3 4𝑥2 − 5

• Answer

• 𝑔′ 𝑥 = 2𝑥 4𝑥2 − 5 − 𝑥2 + 3 8𝑥

Topic 3: 400

• Question:

• Determine the derivative of 𝑓 𝑥 =3𝑥4−5𝑥

𝑥+1

• Answer

• 𝑓′ 𝑥 =𝑥+1 12𝑥3−5 − 3𝑥4−5𝑥

𝑥+1 2

Topic 3: 600

• Question:

• Determine the derivative of 𝑓 𝑥 = 4𝑥2tan𝑥

• Answer

• 𝑓′ 𝑥 = 8𝑥tan𝑥 + 4𝑥2sec2𝑥

Topic 3: 800

• Question:

• Determine the derivative of 𝑔 𝑥 = 𝑥2sin𝑥 − 2sec𝑥

• Answer

• 𝑔′ 𝑥 = 2𝑥sin𝑥 + 𝑥2cos𝑥 − 2sec𝑥tan𝑥

Topic 3: 1000

• Question:

• Determine the derivative of 𝑓 𝑥 =csc𝑥

6𝑥2+𝑥

• Answer

• 𝑓′ 𝑥 =6𝑥2+𝑥 −csc𝑥cot𝑥 −csc𝑥 12𝑥+1

6𝑥2+𝑥 2

Topic 4: 200

• Question:

• Determine the derivative of 𝑦 = 𝑥3 + 2𝑥 37

• Answer

• 𝑦′ = 37 𝑥3 + 2𝑥 36 3𝑥2 + 2

Topic 4: 400

• Question:

• Determine the derivative of 𝑦 = 4 + 3 𝑥

• Answer

• 𝑦′ =1

24 + 3𝑥 Τ1 2

− Τ1 2 3

2𝑥− Τ1 2

Topic 4: 600

• Question:

• Determine the derivative of 𝑓 𝑥 = sin5 3𝑥4

• Answer

• 𝑓′ 𝑥 = 5sin4 3𝑥4 cos 3𝑥4 12𝑥3

Topic 4: 800

• Question:

• Determine the derivative of 𝑦 =3𝑥+2 2

2𝑥3+𝑥 4

• Answer

• y’= 2𝑥3+𝑥

42 3𝑥+2 3 − 3𝑥+2 24 2𝑥3+𝑥

36𝑥2+1

2𝑥3+𝑥 8

Topic 4: 1000

• Question:

• 𝐷𝑥 sin cos sin 2𝑥 =

• Answer

• cos cos sin 2𝑥 −sin sin 2𝑥 cos 2𝑥 2

Topic 5: 200• Question:

• Find the second derivative of

𝑔 𝑥 =4𝑥5 −

7

5𝑥4+ 5𝑥

• Answer

•

𝑔 𝑥 = 𝑥 Τ5 4 −7

5𝑥−4 + 5𝑥

𝑔′(𝑥) =5

4𝑥 Τ1 4 +

28

5𝑥−5 + 5

𝑔′′(𝑥) =5

16𝑥 Τ−3

4 − 28𝑥−6

Topic 5: 400

• Question:

• Find the derivative of 𝑦 = sin6𝑥 − tan 5𝑥3 + 4𝑥

• Answer

• 𝑦′ = (6sin5𝑥)(cos𝑥) − sec2(5𝑥3 + 4𝑥)(15𝑥2 +

Topic 5: 600

• Question:

• Determine the derivative of 𝑔 𝑥 =2

sin2𝑥

• Answer

• 𝑦′ =−4cos𝑥

sin3𝑥or y’ =-4sin-3 x cos x

Topic 5: 800

• Question:

• Determine the derivative of

𝑔 𝑥 =116𝑥5

−3 𝑥

53𝑥2

• Answer

• 𝑔′(𝑥) =−55

6𝑥 Τ−11

6 +1

10𝑥 Τ−7

6

Topic 5: 1000

• Question:

• Find the slope of the tangent line of 𝑓 𝑥 =

cos 5𝑥4 + 6𝑥2 − 2𝑥 at 𝑥 =2𝜋

3

• Answer

•)𝑓′(𝑥) = −sin(5𝑥4 + 6𝑥2 − 2𝑥)(20𝑥3 + 12𝑥 − 2

𝑓′2𝜋

3= 178.811

Bonus Question: 5000 pts.

• Question:

• Find the second derivative of 𝑓 𝑥 =3𝑥−4

7𝑥2+2.

Simplify the first derivative before finding the second derivative.

• Answer

• …

• 𝑓′′ 𝑥 =7𝑥2+2 −42𝑥+56 − −21𝑥2+56𝑥+6 2 14𝑥

7𝑥2+2 3

Daily Double

The Winner Of The Last RoundWrite Down How Much Money

You Are Willing To RiskIf You get the Question write you win that money

If you get it wrong you Loss the money!

Daily Double

The Winner Of The Last RoundWrite Down How Much Money

You Are Willing To RiskIf You get the Question write you win that money

If you get it wrong you Loss the money!