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For more information, please see full course syllabus of College Calculus: Level I
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The Product Rule

  • Memorize the Product Rule – you will be using it a lot!
  • A constant factor can be factored out in front of a derivative.
  • Avoid common errors by remembering to use the Product Rule whenever you are differentiating a product!

The Product Rule

Find the derivative of f(x) = g(x) * h(x) given g(x) = ln(x), g′(x) = [1/x] and h(x) = ex = h′(x)
  • f′(x) = g(x) h′(x) + h(x) g′(x)=
  • Remember, f′(x) is generally equivalent to [d/dx] f(x)
ln(x) ex + ex [1/x]
Find the derivative of y = x sin(x)
  • dy = x [d/dx] sin(x) + sin(x) [d/dx] x
  • = x cos(x) + sin(x) * 1 =
bsection*x cos(x) + sin(x)
Find the derivative of f(x) = sin(x) tan(x)
  • f′(x) = sin(x) [d/dx] tan(x) + tan(x) [d/dx] sin(x)
  • = sin(x) sec2(x) + tan(x) cos(x) =
sin(x) sec2(x) + sin(x)
Show that f′(x) = sin(2x) given f(x) = sin2(x)
  • f′(x) = [d/dx] sin2(x)
  • = [d/dx] (sin(x)sin(x))
  • = sin(x) [d/dx] sin(x) + sin(x) [d/dx] sin(x)
  • = sin(x) cos(x) + sin(x) cos(x)
  • = 2 sin(x) cos(x) =
= sin(2x)
Find the derivative of f(x) = (x3 + x + 3)(x2 + 1)
  • f′(x) = (x3 + x + 3) [d/dx] (x2 + 1) + (x2 + 1) [d/dx] (x3 + x + 3)
  • = (x3 + x + 3) 2x + (x2 + 1)(3x2 + 1)
  • = (2x4 + 2x2 + 6x) + (3x4 + 3x2 + x2 + 1) =
5x4 + 6x2 + 6x + 1
Find the derivative of f(x) = (x + 1)sec(x)
  • f′(x) = (x + 1) [d/dx] sec(x) + sec(x) [d/dx] (x + 1) =
(x + 1) sec(x) tan(x) + sec(x)
Find the derivative of f(x) = (x2 + 4)(x + 3)2
  • f′(x) = (x2 + 4) [d/dx] (x + 3)2 + (x + 3)2 [d/dx] (x2 + 4)
  • = (x2 + 4) [d/dx] (x + 3)2 + (x + 3)2 (2x)
  • = (x2 + 4) [d/dx] (x + 3)2 + (x2 + 6x + 9)(2x)
  • = (x2 + 4) [d/dx] (x + 3)2 + 2x3 + 12x2 + 18x
  • = (x2 + 4) ( (x + 3) [d/dx] (x + 3) + (x + 3) [d/dx] (x + 3) ) + 2x3 + 12x2 + 18x
  • = (x2 + 4) ( (x + 3) + (x + 3) ) + 2x3 + 12x2 + 18x
  • = (x2 + 4) (2x + 6) + 2x3 + 12x2 + 18x
  • = 2x3 + 8x + 6x2 + 24 + 2x3 + 12x2 + 18x
4x3 + 18x2 + 26x + 24
Find the derivative of f(x) = (7x + 53)2
  • f′(x) = (7x + 53) [d/dx] (7x + 53) + (7x + 53) [d/dx] (7x + 53)
14(7x + 53)
Given f(x) = [(x3)/3] + 2x2 + 3x, find the roots of f′(x)
  • The roots are given by values of x such that the function equals zero.
  • f′(x) = [d/dx] ([(x3)/3] + 2x2 + 3x)
  • = x2 + 4x + 3
  • f′(x) = 0 = x2 + 4x + 3
  • = (x + 1) (x + 3) = 0
  • x = −1, −3
  • A LOOK AHEAD: Below is the graph of f(x). Note the behavior of f(x) at the roots of its derivative.
x = −1, −3
Given f(x) = (x + 2)2 + 1, find the roots of f′(x)
  • f′(x) = [d/dx] ( (x + 2)2 + 1 )
  • = [d/dx] (x + 2)2
  • = (x + 2) [d/dx] (x + 2) + (x + 2) [d/dx] (x + 2)
  • = (x + 2) + (x + 2)
  • = 2(x + 2) = 0
  • x = −2
  • Graph of f(x)
x = −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.

Answer

The Product 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.

  • Statement of the Product Rule 0:08
  • Lecture Example 1 0:41
  • Lecture Example 2 2:27
  • Lecture Example 3 5:03
  • Additional Example 4
  • Additional Example 5