1. 3.5 – Derivative of Trigonometric Functions
REVIEW:
Function Notation
𝑦=𝑓(𝑥)
𝑢=𝑓(𝑥)
𝑣=𝑔(𝑥)
𝑠=𝑠(𝑡)
Derivative Notation
𝑦 ′ = 𝑓 ′ 𝑥 = 𝑑𝑦 𝑑𝑥 = 𝑑 𝑑𝑥 𝑓 𝑥 = 𝑑 𝑑𝑥 𝑦= 𝐷 𝑥 𝑓 𝑥 = 𝐷 𝑥 𝑦
𝑢 ′ = 𝑓 ′ 𝑥 = 𝑑𝑢 𝑑𝑥 = 𝑑 𝑑𝑥 𝑓 𝑥 = 𝑑 𝑑𝑥 𝑢= 𝐷 𝑥 𝑓 𝑥 = 𝐷 𝑥 𝑢
𝑣 ′ = 𝑔 ′ 𝑥 = 𝑑𝑣 𝑑𝑥 = 𝑑 𝑑𝑥 𝑔 𝑥 = 𝑑 𝑑𝑥 𝑣= 𝐷 𝑥 𝑓 𝑥 = 𝐷 𝑥 𝑣
𝑠 ′ = 𝑠 ′ 𝑡 = 𝑑𝑠 𝑑𝑡 = 𝑑 𝑑𝑡 𝑠(𝑡) = 𝑑 𝑑𝑡 𝑠= 𝐷 𝑡 𝑠 𝑡 = 𝐷 𝑡 𝑠

2. 3.5 – Derivative of Trigonometric Functions
Derivative of Sine
Derivative of Cosecant
𝑦=𝑓 𝑥 = sin 𝑥
𝑦=𝑓 𝑥 = csc 𝑥
𝑦 ′ = 𝑓 ′ 𝑥 = 𝑑𝑦 𝑑𝑥 = cos 𝑥
𝑦 ′ = 𝑓 ′ 𝑥 = 𝑑𝑦 𝑑𝑥 =−𝑐𝑜𝑡𝑥 𝑐𝑠𝑐𝑥
Derivative of Cosine
Derivative of Secant
𝑦=𝑓 𝑥 = cos 𝑥
𝑦=𝑓 𝑥 = sec 𝑥
𝑦′=𝑓′ 𝑥 = 𝑑𝑦 𝑑𝑥 =− sin 𝑥
𝑦′=𝑓′ 𝑥 = 𝑑𝑦 𝑑𝑥 = tan 𝑥 sec 𝑥
Derivative of Tangent
Derivative of Cotangent
𝑦=𝑓 𝑥 = tan 𝑥
𝑦=𝑓 𝑥 = cot 𝑥
𝑦′=𝑓′ 𝑥 = 𝑑𝑦 𝑑𝑥 = 𝑠𝑒𝑐 2 𝑥
𝑦′=𝑓′ 𝑥 = 𝑑𝑦 𝑑𝑥 =− 𝑐𝑠𝑐 2 𝑥

3. 3.5 – Derivative of Trigonometric Functions

4. 3.5 – Derivative of Trigonometric Functions

5. 3.5 – Derivative of Trigonometric Functions
Practice Problems – Worksheet
Derivatives of Trigonometric Functions

7. 3.6 – The Chain Rule Review of the Product Rule: 𝑦= 3 𝑥 3 +2 𝑥 2 2
𝑦= 3 𝑥 3 +2 𝑥 2 2
= 3 𝑥 3 +2 𝑥 𝑥 3 +2 𝑥 2
𝑦′= 3 𝑥 3 +2 𝑥 𝑥 2 +4𝑥 + 9 𝑥 2 +4𝑥 3 𝑥 3 +2 𝑥 2
𝑦′=2 3 𝑥 3 +2 𝑥 𝑥 2 +4𝑥
𝑦= 6 𝑥 2 +𝑥 3
= 6 𝑥 2 +𝑥 6 𝑥 2 +𝑥 6 𝑥 2 +𝑥
𝑦 ′ = 6 𝑥 2 +𝑥 6 𝑥 2 +𝑥 12𝑥 𝑥 2 +𝑥 12𝑥+1 6 𝑥 2 +𝑥 + 12𝑥+1 6 𝑥 2 +𝑥 6 𝑥 2 +𝑥
𝑦 ′ = 6 𝑥 2 +𝑥 𝑥 𝑥 2 +𝑥 𝑥 𝑥 2 +𝑥 𝑥+1
𝑦′=3 6 𝑥 2 +𝑥 𝑥+1

8. 3.6 – The Chain Rule 𝑦= 3 𝑥 3 +2 𝑥 2 2 𝑦′=2 3 𝑥 3 +2 𝑥 2 9 𝑥 2 +4𝑥
𝑦= 3 𝑥 3 +2 𝑥 2 2
𝑦′=2 3 𝑥 3 +2 𝑥 𝑥 2 +4𝑥
𝑦= 6 𝑥 2 +𝑥 3
𝑦′=3 6 𝑥 2 +𝑥 𝑥+1
Additional Problems:
𝑦= 𝑥 3 +2𝑥 9
𝑦′= 9
𝑥 3 +2𝑥 8
3 𝑥 2 +2
𝑦= 5𝑥
𝑦′= 4 5𝑥
10𝑥
𝑦= 2𝑥 5 −3 𝑥 4 +𝑥−3 13
𝑦′= 13
2𝑥 5 −3 𝑥 4 +𝑥−3 12
10𝑥 4 −12 𝑥 3 +1

9. Practice Problems – Worksheet
3.6 – The Chain Rule
Practice Problems – Worksheet
The Chain Rule