INSTRUCTORS Raffi Hovasapian John Zhu

Start learning today, and be successful in your academic & professional career. Start Today!

• ## Related Books

### Chain Rule

• Or,
• Don’t forget to multiply after evaluating (common hasty error)

### Chain Rule

Find f′(x) if f(x) = (3x5 + 7x + 11)13
• u(x) = 3x5 + 7x + 11
• [du/dx] = u′(x) = 15x4 + 7
• f(x) = u13
• f′(x) = [d/dx] u13
• f′(x) = 13 u12 [du/dx]
13(3x5 + 7x + 11)12 (15x4 + 7)
Find f′(x) if f(x) = √{x3 + 5x}
• u = x3 + 5x
• [du/dx] = 3x2 + 5
• f′(x) = [d/dx] √u
• f′(x) = [d/dx] u[1/2]
• = [1/2] (u−[1/2]) [du/dx]
• = [du/(2√u)] =
[(3x2 + 5)/(2√{x3 + 5x})]
Find f′(x) if f(x) = sin2(x)
• u = sin(x)
• u′ = cos(x)
• f′(x) = [d/dx] u2
• = 2 u u′
• = 2 sin(x) cos(x) =
sin(2x)
Find f′(x) if f(x) = (sec(5x))9
• u = sec(5x), v = 5x, v′ = 5
• u = sec(v)
• u′ = sec(v)tan(v) v′
• u′ = 5sec(5x)tan(5x)
• f′(x) = [d/dx] u9
• = 9 u8 u′
• = 9 (sec(5x))8 (5 sec(5x) tan(5x)) =
45 tan(5x) (sec(5x))9
Find f′(x) if f(x) = cot(5x2 + 3x)
• f′(x) = −csc2(5x2 + 3x) [d/dx] (5x2 + 3x)
• = −csc2(5x2 + 3x) (10x + 3) =
−(10x + 3)csc2(5x2 + 3x)
Find f′(x) if f(x) = [(x2)/tan(3x)]
• g = x2, g′ = 2x
• h = tan(3x), h′ = sec2(3x) [d/dx] 3x = 3 sec2(3x)
• f′(x) = [d/dx] [g/h]
• = [(h g′− g h′)/(h2)]
• = [(tan(3x) (2x) − x2 (3 sec2(3x)))/(tan2(3x))] =
[(2x tan(3x) − 3x2 sec2(3x))/(tan2(3x))]
Find f′(x) if f(x) = (x2 + 8x + 16)(x + 4)3
• f(x) = (x2 + 8x + 16) (x + 4)3
• = (x + 4)2 (x + 4)3
• = (x + 4)5
• f′(x) = 5 (x + 4)4 [d/dx] (x + 4)
5 (x + 4)4
Find f′(x) if f(x) = (7x9 + √{3x})(x + 1)4
• u = 7x9 + √{3x}, u′ = 63x8 + [(√3)/(2√x)]
• v = (x + 1)4, v′ = 4 (x + 1)3 [d/dx] (x + 1) = 4 (x + 1)3
• f′(x) = [d/dx] u v
• = u v′+ v u′=
(7x9 + √{3x})(4 (x + 1)3) + (x + 1)4 (63x8 + [(√3)/(2√x)])
Find f′(x) if f(x) = sin(sin(x))
• u = sin(x), u′ = cos(x)
• f′(x) = [d/dx] sin(u)
• = cos(u) u′=
cos(sin(x)) cos(x)
Find f′(x) if f(x) = √{csc(11x)}
• u = csc(11x), v = 11x, v′ = 11
• u = csc(v), u′ = −cot(v)csc(v) v′
• u′ = −11 cot(11x)csc(11x)
• f′(x) = [d/dx] u[1/2]
• = [1/(2√u)] u′
• = [(−11 cot(11x)csc(11x))/(2√{csc(11x)})] =
[(−11cot(11x)√{csc(11x)})/2]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Chain Rule

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Chain Rule 0:07
• Definition
• Example 1: Applying the Chain Rule
• Example 2 4:25
• Example 3 6:02
• Example 4 9:25
• Example 5 12:47
• Example 6 15:27