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INSTRUCTORS Raffi Hovasapian John Zhu
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For more information, please see full course syllabus of Calculus AB
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Lecture Comments (2)

0 answers

Post by Kushal Patel on October 7, 2013

explain why and how ?

0 answers

Post by GEORGE KAMAU on June 3, 2013

Do you mean differeniable over x axis or Y axis on your last example

Derivative Definition & Properties

  • Definitions
    • Approximation of slope

Derivative Definition & Properties

Find the slope of f(x) = x2 − 4 at x = 1
  • f′(x) = limh → 0 [(f(x + h) − f(x))/h]
  • = limh → 0 [((x + h)2 − 4 − (x2 − 4))/h]
  • = limh → 0 [(x2 + 2xh + h2 − 4 − x2 + 4)/h]
  • = limh → 0 [(2xh + h2)/h]
  • = limh → 0 2x + h
  • f′(x) = 2x
  • f′(1) = 2(1) =
2
Find the slope of f(x) = x2 −4 at x = 10
  • f′(x) = 2x
  • f′(10) = 2(10) =
20
Find the derivative of f(x) = x2 + 1024
  • f′(x) = limh → 0 [((x + h)2 + 1024 − (x2 + 1024))/h]
  • = limh → 0 [(x2 + 2xh + h2 + 1024 − x2 − 1024)/h]
  • = limh → 0 [(2xh + h2)/h]
  • = limh → 0 2x + h
  • f′(x) =
2x
Find the slope of f(x) = 1 − [(x2)/2] at x = 1
  • f′(x) = limh → 0 [(1 − [((x + h)2)/2] − 1 + [(x2)/2])/h]
  • = limh → 0 [(−[(x2 + 2xh + h2)/2] + [(x2)/2])/h]
  • = limh → 0 [(−xh − [(h2)/2])/h]
  • = limh → 0 −[xh/h]
  • = −x limh → 0 [h/h]
  • f′(x) = −x
f′(1) = −1
Find the derivative of f(x) = sin(x)
  • f′(x) = limh → 0 [(sin(x + h) − sin(x))/h]
  • = limh → 0 [(sin(x)cos(h) + cos(x)sin(h) − sin(x))/h]
  • = limh → 0 [(sin(x)cos(h) − sin(x))/h] + limh → 0 [cos(x)sin(h)/h]
  • = sin(x)limh → 0 [(cos(h) − 1)/h] + cos(x) limh → 0 [sin(h)/h]
  • = sin(x) * 0 + cos(x) * 1 =
cos(x)
Find the slope of f(x) = sin(x) at x = π
  • f′(x) = cos(x)
  • f′(π) = cos(π) =
−1
Find the derivative of f(x) = cos(x)
  • f′(x) = limh → 0 [(cos(x + h) − cos(x))/h]
  • = limh → 0 [(cos(x)cos(h) − sin(x)sin(h) − cos(x))/h]
  • = limh → 0 [(cos(x)cos(h) − cos(x))/h] − limh → 0 [sin(x)sin(h)/h]
  • = cos(x) limh → 0 [(cos(h) − 1)/h] − sin(x) limh → 0 [sin(h)/h]
  • = cos(x) * 0 − sin(x) * 1 =
−sin(x)
Find the slope of f(x) = cos(x) at x = π
  • f′(x) = −sin(x)
  • f′(π) = −sin(π) =
0
Find the slope off(x) = |x + 4| at x = 0
  • f(x) = |x + 4|
  • For , we only need to worry about the top equation as zero is greater than -4
0Find the slope of at
  • The slope changes instantaneously from -1 to 1 when approaching -4. There is a "corner" at .
The function is not differentiable at .

*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

Derivative Definition & Properties

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
  • Definition 0:09
    • Formal Definition
    • Difference Quotient
  • Basic Derivatives 1:16
  • Differentiability 2:54