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

• ## Related Books

### The Quotient Rule

• Memorize the Quotient Rule – you will be using it a lot!
• A constant factor can be factored out in front of a derivative.
• A function of the form can be re-written as if you prefer the Chain Rule to the Quotient Rule.
• Avoid common errors by remembering to use the Quotient Rule whenever you are differentiating a quotient!

### The Quotient Rule

Given f(x) = 3x and g(x) = 4x, find [d/dx] [f(x)/g(x)]
• [d/dx] [f(x)/g(x)] = [(g(x)f′(x) − f(x)g′(x))/(g(x)2)]
• = [(4x [d/dx] 3x − 3x [d/dx] 4x)/((4x)2)]
• = [(12x − 12x)/(16x2)]
0
Find f′(x) if f(x) = [3x/(x2 + 1)]
• f′(x) = [3x/(x2 + 1)]
• = [((x2 + 1) [d/dx] 3x − 3x [d/dx] (x2 + 1))/((x2 + 1)2)]
• = [((x2 + 1)3 − 3x(2x))/((x2 + 1)2)]
• = [(3x2 −6x2 + 3)/((x2 + 1)2)]
[(−3 (x2 − 1))/((x2 + 1)2)]
Find the derivative of tan(x) using the quotient rule
• [d/dx] tan(x) = [d/dx] [sin(x)/cos(x)]
• = [(cos(x) [d/dx] sin(x) − sin(x) [d/dx] cos(x))/(cos2(x))]
• = [(cos(x)cos(x) − sin(x) (−sin(x)))/(cos2(x))]
• = [(cos2(x) + sin2(x))/(cos2(x))]
• = [1/(cos2(x))]
sec2(x)
Find the derivative of cot(x) using the quotient rule
• [d/dx] cot(x) = [d/dx] [cos(x)/sin(x)]
• = [(sin(x) [d/dx] cos(x) − cos(x) [d/dx] sin(x))/(sin2(x))]
• = [(sin(x) (−sin(x)) − cos(x) cos(x))/(sin2(x))]
• = −[(sin2(x) + cos2(x))/(sin2(x))]
• = −[1/(sin2(x))]
−csc2(x)
Find the derivative of sec(x) using the quotient rule
• [d/dx] sec(x) = [d/dx] [1/cos(x)]
• = [(cos(x) [d/dx] 1 − 1 [d/dx] cos(x))/(cos2(x))]
• = [(0 − (−sin(x)))/(cos2(x))]
• = [sin(x)/(cos2(x))]
• = sec(x) [sin(x)/cos(x)]
sec(x) tan(x)
Find f′(x) if f(x) = [(x3)/((5x + 4)tan(x))]
• f′(x) = [((5x + 4) tan(x) [d/dx] x3 − x3 [d/dx] ((5x + 4) tan(x)t))/(t((5x + 4)tan(x))2)]
• = [((5x + 4) tan(x) (3x2) − x3 [d/dx] ((5x + 4) tan(x)))/((5x + 4)2 tan2(x))]
• = [((5x + 4) tan(x) (3x2) − x3 ( (5x + 4) [d/dx] tan(x) + tan(x) [d/dx] (5x + 4) ))/((5x + 4)2 tan2(x))]
• = [((5x + 4) tan(x) (3x2) − x3 ( (5x + 4) sec2(x) + tan(x) 5 ))/((5x + 4)2 tan2(x))]
[((5x + 4) tan(x) (3x2) − x3 ( (5x + 4) sec2(x) + 5tan(x)))/((5x + 4)2 tan2(x))]
Find f′(x) if f(x) = [(√x + 3)/(x4 −16)]
• f′(x) = [(√x + 3)/(x4 − 16)]
• = [((x4 − 16) [d/dx] (√x + 3) − (√x + 3) [d/dx] (x4 − 16))/((x4 − 16)2)]
[((x4 − 16) [1/2] x−[1/2] − (√x + 3) (4x3))/((x4 − 16)2)]
Find f′(x) if f(x) = [(.2x6)/(.1x + cos(x))]
• f′(x) = [((.1x + cos(x)) [d/dx] .2x6 − (.2x6) [d/dx] (.1x + cos(x)))/((.1x + cos(x))2)]
[((.1x + cos(x)) 1.2x5 − .2x6 (.1 − sin(x)))/((.1x + cos(x))2)]
Find f′(t) if f(t) = [(t2)/sin(t)]
• The letter or symbol used for the variable is not important. What's important is consistency.
• f′(t) = [(sin(t) [d/dt] t2 − t2 [d/dt] sin(t))/(sin2(t))]
• = [(sin(t) (2t) − t2 cos(t))/(sin2(t))]
[(t(2sin(t) − t cos(t)))/(sin2(t))]
Find f′(1) if f(z) = [(z − 3)/(2 − z)]
• f′(t) = [((2 − z) [d/dz] (z − 3) − (z − 3) [d/dz] (2 − z))/((2 − z)2)]
• = [((2 − z) 1 − (z − 3) (−1))/((2 − z)2)]
• = [(2 − z + z − 3)/((2 − z)2)] =
−[1/((2 − z)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.

### The Quotient 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
• Statement of the Quotient Rule 0:07
• Carrying out the Differentiation
• Quotient Rule in Words
• Lecture Example 1 1:19
• Lecture Example 2 4:23
• Lecture Example 3 8:00