University:
Georgia Southern University
Course:
MATH 1441 | Calculus I
Academic year:

2024
Views:

184

Pages:

1
Author:

Jalen W.

The Chain Rule (The derivative of a Composite Function) Without the Chain Rule With the Chain Rule 𝑦 = 𝑥2 + 1 𝑦 = √𝑥 2 + 1 𝑦 = 5𝑥 5 − 3𝑥 + 1 𝑦= 𝑦= 𝑦 = (3𝑥 2 + 1)10 1 𝑥 𝑦= 1 𝑥5 𝑦= 𝑦= 𝑥+1 𝑥−1 𝑥3 1 + 2𝑥 (𝑥 2 4 𝑦=√ 1 + 𝑥 + 1)5 𝑥+1 𝑥−1 Rule 8: The Chain Rule If 𝑦 = 𝑓(𝑢) is a differentiable function of u and 𝑢 = 𝑔(𝑥) is a differentiable function of 𝑥, then 𝑦 = 𝑓(𝑔(𝑥)) is a differentiable function of 𝑥 and Differentiate with respect to 𝑥. 1. 𝑦 = 𝑒 cos 𝑥 2. 𝑦 = (5𝑥 3 − 𝑥 4 )7 3. 𝑦 = 𝑠𝑖𝑛5 𝑥 4. 𝑦 = 𝑠𝑖𝑛3 (4𝑥) 5. 𝑦 = 𝑒 √3𝑥+1 6. 𝑟 = sin(𝜃 2 ) 7. 𝑟 = tan(4𝜃) sec(4𝜃) 8. 𝑟 = sin(𝜃 2 ) cos(2𝜃) 𝑑 [𝑓(𝑔(𝑥))] 𝑑𝑥 = 𝑓 ′ (𝑔(𝑥)) 𝑔′ (𝑥).