INSTRUCTORS Raffi Hovasapian John Zhu

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For more information, please see full course syllabus of Calculus AB

• ## Related Books

 0 answersPost by Alex Moon on March 31, 2014The sound only plays on my left headphone speaker. Perhaps its not in a dual audio format? 1 answerLast reply by: Arshin JainMon Oct 21, 2013 9:24 PMPost by GEORGE KAMAU on June 3, 2013I think your sound is on mute.. can view the video but can here the sound. All my settings are ok tho.. 1 answerLast reply by: Ash NiaziFri Dec 28, 2012 2:35 PMPost by Alyssa Snow on October 14, 2012On example three, shouldn't the final line on the denominator read e^2, instead of e^2x?

### Derivatives of Exponential Functions

• All other derivative rules still apply
• uis usually substituted with expression including x

### Derivatives of Exponential Functions

Find f′(x) if f(x) = 10g(x)
• f′(x) = [d/dx] 10g(x) =
10g(x) ln(10) g′(x)
Find f′(x) if f(x) = eg(x)
• f′(x) = [d/dx] eg(x)
• = eg(x) (lne) g′(x)
• = eg(x) (1) g′(x) =
eg(x) g′(x)
Given f(x) = (1 + x + [(x2)/2!] + [(x3)/3!] + [(x4)/4!] + ...), find f′(x)
• f′(x) = [d/dx] (1 + x + [(x2)/2!] + [(x3)/3!] + [(x4)/4!] + ...) =
• 1 + x + [(x2)/2!] + [(x3)/3!] + ...
• One can extrapolate that f(x) = f′(x) in this case. It just so happens that f(x) is the taylor series expansion of ex
• ex = 1 + x + [(x2)/2!] + [(x3)/3!] + [(x4)/4!] + ...
1 + x + [(x2)/2!] + [(x3)/3!] + ...
Find f′(t) if f(t) = e2t sin(t)
• f′(t) = [d/dt] e2t sint
• = e2t [d/dt] sint + sint [d/dt] e2t
• = e2t cost + sint (e2t) [d/dt] 2t
• = e2t cost + (sint) e2t 2 =
e2t ( cost + 2 sint )
Find f′(θ) if f(θ) = etan(θ)
• f′(θ) = [d/(d θ)] etan(θ)
• = etan(θ) [d/(d θ)] tan(θ) =
etan(θ) sec2(θ)
Find f′(x) if f(x) = [(e2x)/(ex + 1)]
• f′(x) = [d/dx] [(e2x)/(ex + 1)]
• = [((ex + 1) [d/dx] e2x − e2x [d/dx] (ex + 1))/((ex + 1)2)]
• = [((ex + 1) e2x [d/dx] (2x) − e2x ex [d/dx] x)/((ex + 1)2)]
• = [((ex + 1) 2 e2x − e2x ex)/((ex + 1)2)]
• = [(e2x(2ex + 2 − ex))/((ex + 1)2)] =
[(e2x(ex + 2))/((ex + 1)2)]
Find f′(x) if f(x) = eex
• f′(x) = [d/dx] eex
• = eex [d/dx] ex
• = ex eex =
ex + ex
Find f"(x) if f(x) = sin(e−x)
• f′(x) = [d/dx] sin(e−x)
• = cos(e−x) [d/dx] e−x
• = cos(e−x) (−e−x)
• = −e−x cos(e−x)
• f"(x) = [d/dx] (−e−x cos(e−x))
• = −e−x [d/dx] cos(e−x) + (−cos(e−x) [d/dx] e−x)
• = −e−x (−sin(e−x) [d/dx] e−x) − cos(e−x) (−e−x)
• = e−x sin(e−x) (−e−x) + cos(e−x) e−x =
−e−2x sin(e−x) + e−x cos(e−x)
Find the second derivative of f(x) = ex2
• f′(x) = [d/dx] ex2
• = ex2 [d/dx] x2
• = 2x ex2
• f"(x) = 2x [d/dx] ex2 + ex2 [d/dx] (2x)
• = 2x(2x ex2) + 2 ex2
• = 4x2 ex2 + 2 ex2 =
2ex2(2x2 + 1)
Find f′(x) if f(x) = ex2 + sin(x) + 2
• u = x2 + sinx + 2, u′ = 2x + cosx
• f′(x) = [d/dx] eu
• = eu u′=
ex2 + sin(x) + 2(2x + cosx)

*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.

### Derivatives of Exponential Functions

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
• Types of Derivatives: Exponential Functions 0:08
• Derivatives: Definition/ Formula
• Example 1
• Example 2 2:47
• Example 3 4:13
• Example 4 7:11
• Example 5 9:23
• Example 6 11:06