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Lecture Comments (5)

0 answers

Post by Alex Moon on March 31, 2014

The sound only plays on my left headphone speaker. Perhaps its not in a dual audio format?

1 answer

Last reply by: Arshin Jain
Mon Oct 21, 2013 9:24 PM

Post by GEORGE KAMAU on June 3, 2013

I think your sound is on mute.. can view the video but can here the sound. All my settings are ok tho..

1 answer

Last reply by: Ash Niazi
Fri Dec 28, 2012 2:35 PM

Post by Alyssa Snow on October 14, 2012

On 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
  • u is 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.

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