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, start superscript, 2, end superscript, 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, start superscript, 2, end superscript.
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, start superscript, 2, end superscript, 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, start superscript, 2, end superscript, 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, start superscript, 2, end superscript, 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, start superscript, 2, end superscript, 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, p, i or 2, slash, 3, space, p, i

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