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:
ddx[f(x)g(x)]=ddx[f(x)]g(x)f(x)ddx[g(x)][g(x)]2\dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right]=\dfrac{\dfrac{d}{dx}[f(x)]\cdot g(x)-f(x)\cdot\dfrac{d}{dx}[g(x)]}{[g(x)]^2}
Basically, you take the derivative of ff multiplied by gg, subtract ff multiplied by the derivative of gg, and divide all that by [g(x)]2[g(x)]^2.
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 sin(x)x2\dfrac{\sin(x)}{x^2}:
=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(x)=x2exf(x)=\dfrac{x^2}{e^x}
f(x)=f'(x)=

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

Example 2

Suppose we are given this table of values:
xxf(x)f(x)g(x)g(x)f(x)f'(x)g(x)g'(x)
444-42-20088
H(x)H(x) is defined as f(x)g(x)\dfrac{f(x)}{g(x)}, and we are asked to find H(4)H'(4).
The Quotient rule tells us that H(x)H'(x) is f(x)g(x)f(x)g(x)[g(x)]2\dfrac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}. This means H(4)H'(4) is f(4)g(4)f(4)g(4)[g(4)]2\dfrac{f'(4)g(4)-f(4)g'(4)}{[g(4)]^2}. 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
xxg(x)g(x)h(x)h(x)g(x)g'(x)h(x)h'(x)
2-244111-122
F(x)=g(x)h(x)F(x)=\dfrac{g(x)}{h(x)}
F(2)=F'(-2)=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

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