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Quotient rule review

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.B (LO)
,
FUN‑3.B.2 (EK)
Review your knowledge of the Quotient rule for derivatives, and use it to solve problems.

What is the Quotient rule?

The Quotient rule tells us how to differentiate expressions that are the quotient of two other, more basic, expressions:
start fraction, d, divided by, d, x, end fraction, open bracket, start fraction, f, left parenthesis, x, right parenthesis, divided by, g, left parenthesis, x, right parenthesis, end fraction, close bracket, equals, start fraction, start fraction, d, divided by, d, x, end fraction, open bracket, f, left parenthesis, x, right parenthesis, close bracket, dot, g, left parenthesis, x, right parenthesis, minus, f, left parenthesis, x, right parenthesis, dot, start fraction, d, divided by, d, x, end fraction, open bracket, g, left parenthesis, x, right parenthesis, close bracket, divided by, open bracket, g, left parenthesis, x, right parenthesis, close bracket, squared, end fraction
Basically, you take the derivative of f multiplied by g, subtract f multiplied by the derivative of g, and divide all that by open bracket, g, left parenthesis, x, right parenthesis, close bracket, squared.
Want to learn more about the Quotient rule? Check out this video.

What problems can I solve with the Quotient rule?

Example 1

Consider the following differentiation of start fraction, sine, left parenthesis, x, right parenthesis, divided by, x, squared, end fraction:
=ddx(sin(x)x2)=ddx(sin(x))x2sin(x)ddx(x2)(x2)2Quotient rule=cos(x)x2sin(x)2x(x2)2Differentiate sin(x) and x2=x(xcos(x)2sin(x))x4Simplify=xcos(x)2sin(x)x3Cancel common factors\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(\dfrac{\sin(x)}{x^2}\right) \\\\ &=\dfrac{\dfrac{d}{dx}(\sin(x))x^2-\sin(x)\dfrac{d}{dx}(x^2)}{(x^2)^2}&&\gray{\text{Quotient rule}} \\\\ &=\dfrac{\cos(x)\cdot x^2-\sin(x)\cdot 2x}{(x^2)^2}&&\gray{\text{Differentiate }\sin(x)\text{ and }x^2} \\\\ &=\dfrac{x\left(x\cos(x)-2\sin(x)\right)}{x^4}&&\gray{\text{Simplify}} \\\\ &=\dfrac{x\cos(x)-2\sin(x)}{x^3}&&\gray{\text{Cancel common factors}} \end{aligned}

Check your understanding

Problem 1
f, left parenthesis, x, right parenthesis, equals, start fraction, x, squared, divided by, e, start superscript, x, end superscript, end fraction
f, prime, left parenthesis, x, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

Example 2

Suppose we are given this table of values:
xf, left parenthesis, x, right parenthesisg, left parenthesis, x, right parenthesisf, prime, left parenthesis, x, right parenthesisg, prime, left parenthesis, x, right parenthesis
4minus, 4minus, 208
H, left parenthesis, x, right parenthesis is defined as start fraction, f, left parenthesis, x, right parenthesis, divided by, g, left parenthesis, x, right parenthesis, end fraction, and we are asked to find H, prime, left parenthesis, 4, right parenthesis.
The Quotient rule tells us that H, prime, left parenthesis, x, right parenthesis is start fraction, f, prime, left parenthesis, x, right parenthesis, g, left parenthesis, x, right parenthesis, minus, f, left parenthesis, x, right parenthesis, g, prime, left parenthesis, x, right parenthesis, divided by, open bracket, g, left parenthesis, x, right parenthesis, close bracket, squared, end fraction. This means H, prime, left parenthesis, 4, right parenthesis is start fraction, f, prime, left parenthesis, 4, right parenthesis, g, left parenthesis, 4, right parenthesis, minus, f, left parenthesis, 4, right parenthesis, g, prime, left parenthesis, 4, right parenthesis, divided by, open bracket, g, left parenthesis, 4, right parenthesis, close bracket, squared, end fraction. Now let's plug the values from the table in the expression:
H(4)=f(4)g(4)f(4)g(4)[g(4)]2=(0)(2)(4)(8)(2)2=324=8\begin{aligned} H'(4)&=\dfrac{f'(4)g(4)-f(4)g'(4)}{[g(4)]^2} \\\\ &=\dfrac{(0)(-2)-(-4)(8)}{(-2)^2} \\\\ &=\dfrac{32}{4} \\\\ &=8 \end{aligned}

Check your understanding

Problem 1
xg, left parenthesis, x, right parenthesish, left parenthesis, x, right parenthesisg, prime, left parenthesis, x, right parenthesish, prime, left parenthesis, x, right parenthesis
minus, 241minus, 12
F, left parenthesis, x, right parenthesis, equals, start fraction, g, left parenthesis, x, right parenthesis, divided by, h, left parenthesis, x, right parenthesis, end fraction
F, prime, left parenthesis, minus, 2, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Want to try more problems like this? Check out this exercise.

Want to join the conversation?

  • blobby green style avatar for user J Kuijpers
    In my studybook from college I've found this problem:
    differentiate ""ln((1+e^x)/(1+e^(-x)))""
    and I don't seem to be able to crack it, how kan I solve this one?
    (13 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      There's even a quicker way! We can first simplify the expression inside the logarithm, by factoring out e^x in the numerator:
      (1+e^x)/(1+e^(-x)) = (e^x)(e^(-x)+1)/(1+e^(-x)) = (e^x)(1+e^(-x))/(1+e^(-x)) = e^x.
      Therefore, we have
      d/dx of (ln((1+e^x)/(1+e^(-x)))) = d/dx of (ln(e^x)) = d/dx of (x) = 1. Amazing!
      (46 votes)
  • spunky sam blue style avatar for user George Emmanuel
    should it not be written like this [d/dx(g(x)) .f(x)-
    d/d(x)(f(x)).g(x)]/[g(x)]^2, i found this lesson extremely baffling as this opposes what i learned from my textbook, which formula is the right one? any suggestion to improve my understanding,please?
    (1 vote)
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  • blobby green style avatar for user Angelo Luidens
    In the first Checking Your Understanding above, isn't the d/dx of e^x = xe^(x-1) ? Else what is the rule applied to e^x?
    (3 votes)
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  • blobby green style avatar for user Scott Thomas Carter
    What would be the reason that one couldn't simply take a quotient (for example (x^2 - 3)/x^4 and simply make it the product (x^2-3)*x^-4 and then apply the product rule? Of course, they have different answers because the product rule does not factor in the square of the denominator, but what rule would be broken by doing so?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      Yes, you can express (x^2 - 3)/x^4 as the product (x^2 - 3) * x^-4 and use the product rule to take the derivative. No rule is broken here. Your answer might not appear the same as if you used the quotient rule to differentiate (x^2 - 3)/x^4, but it should end up mathematically equivalent. If not, then you've made an algebraic or differentiation error somewhere.

      In fact, the product and chain rules can be used to derive the quotient rule:

      d/dx of [f(x)/g(x)] = d/dx of {f(x)*[g(x)]^(-1)}
      = f(x)*(-1)[g(x)]^(-2)*g'(x) + f'(x)*[g(x)]^(-1)
      = [g(x)]^(-2) * [-f(x)*g'(x) + f'(x)*g(x)]
      = [g(x)*f'(x) - f(x)*g'(x)]/[g(x)]^2.

      Have a blessed, wonderful day!
      (2 votes)
  • piceratops tree style avatar for user Daleks Incorporated
    For the AP test, do you need to fully simplify your answers like you require in the review questions listed here?
    (2 votes)
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  • blobby green style avatar for user mwood21
    For question 1 I used x^2 as my f(x) instead of e^x and ended up getting [e^x(2-x)]/(x^3). Would this be incorrect?
    (1 vote)
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    • piceratops ultimate style avatar for user Hecretary Bird
      Division isn't commutative like multiplication, so if you switch the positions of the numbers you're dividing, you'll get a different answer. From this, it follows that the derivative of one function divided by a second one would be different than the derivative of the second divided by the first. You don't have to be careful about this when doing the product rule, but when doing the quotient rule, remember that you subtract term with the derivative of the bottom function, and divide by the bottom function squared. Anything else shouldn't give you the right answer, and (e^x (2-x)) / x^3 would be incorrect.
      (2 votes)
  • male robot hal style avatar for user kevinclauson99
    In problem 2 of example 1 you state that the answer is

    (−​xsin(x)+3cos(x))/x^4 however the actual answer is (which the site recognizes) is flip out that middle "+" for a "-"
    ​​
    (1 vote)
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  • blobby green style avatar for user kris kirk
    so if i had for example '''y=200x/3+5x^2''' and had to differentiate with respect to y, how would i go about this? ( im pretty certain it's quotient rule) i THINK i have it but i also think i've done it with respect to x though...
    (1 vote)
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    • piceratops ultimate style avatar for user ggadget6
      No quotient rule required :). You just need the normal derivative rules. Since there are no x's in the denominator, only constants, you can treat 200/3 as a constant, and just use the normal power rule. In this case, your answer would be dy/dx = 200/3 + 10x.
      (1 vote)
  • blobby green style avatar for user muhammad muttayab
    can rational function be differentiated by doing partial function ? if yes,then which one is more preferable partial fraction or quotient rule?
    (1 vote)
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  • blobby green style avatar for user kaitlynmarushia95
    5x^2/x^2 -9

    Why is the 5 put to the side when getting f'(x)
    the answer I continue to get is 10x(10x^2 +10x-90)/ (x^2 -9)^2

    But the correct answer is -90x/(x^2 -9)^2

    I hope someone can help explain this to me.
    (1 vote)
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    • aqualine sapling style avatar for user Timur Kozhamuratov
      If we set 5x^2 = g(x) (for example)
      and x^2-9 = h(x)(for example)
      g'(x) = 5 (2x^2-1)= 10x
      h'(x) = 2x (because h'(x)= 2x^2-1 and 9 =0 according to "power rule",
      which you can check out on Khan academy).
      then f'(x)=g'(x)/h'(x)
      = (g'(x)*h(x)-g(x)*h'(x))/g(x)^2
      substitute (10x)(x^2-9)-(5x^2)(2x)/(x^2-9)^2
      = 10x^3-90x-10x^3/(x^2-9)^2
      = -90x/(x^2-9)^2
      Hope that helps...;)
      (1 vote)