Quotient Rule — Practice Problems and Worked Solutions

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16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Differentiate using the quotient rule.

   $$f(x)=\frac{x}{x+1}$$

   Answer: $\frac{1}{{\left(x+1\right)}^2}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=1,h^{\prime }(x)=1$$
   3. $$f^{\prime }(x)=\frac{1}{{\left(x+1\right)}^2}$$
2. 2.

   Differentiate using the quotient rule.

   $$f(x)=\frac{x^2}{x+1}$$

   Answer: $\frac{x\left(x+2\right)}{x^2+2x+1}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=2x,h^{\prime }(x)=1$$
   3. $$f^{\prime }(x)=\frac{x\left(x+2\right)}{x^2+2x+1}$$
3. 3.

   Differentiate using the quotient rule.

   $$f(x)=\frac{2x}{x^2+1}$$

   Answer: $\frac{2\left(1-x^2\right)}{x^4+2x^2+1}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=2,h^{\prime }(x)=2x$$
   3. $$f^{\prime }(x)=\frac{2\left(1-x^2\right)}{x^4+2x^2+1}$$
4. 4.

   Differentiate using the quotient rule.

   $$f(x)=\frac{x+1}{x-1}$$

   Answer: $-\frac{2}{{\left(x-1\right)}^2}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=1,h^{\prime }(x)=1$$
   3. $$f^{\prime }(x)=-\frac{2}{{\left(x-1\right)}^2}$$
5. 5.

   Differentiate using the quotient rule.

   $$f(x)=\frac{x^2+1}{x}$$

   Answer: $1-\frac{1}{x^2}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=2x,h^{\prime }(x)=1$$
   3. $$f^{\prime }(x)=1-\frac{1}{x^2}$$
6. 6.

   Differentiate using the quotient rule.

   $$f(x)=\frac{x-2}{x+2}$$

   Answer: $\frac{4}{{\left(x+2\right)}^2}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=1,h^{\prime }(x)=1$$
   3. $$f^{\prime }(x)=\frac{4}{{\left(x+2\right)}^2}$$
7. 7.

   Differentiate using the quotient rule.

   $$f(x)=\frac{x}{x+3}$$

   Answer: $\frac{3}{{\left(x+3\right)}^2}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=1,h^{\prime }(x)=1$$
   3. $$f^{\prime }(x)=\frac{3}{{\left(x+3\right)}^2}$$
8. 8.

   Differentiate using the quotient rule.

   $$f(x)=\frac{x^2}{x-1}$$

   Answer: $\frac{x\left(x-2\right)}{x^2-2x+1}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=2x,h^{\prime }(x)=1$$
   3. $$f^{\prime }(x)=\frac{x\left(x-2\right)}{x^2-2x+1}$$
9. 9.

   Differentiate using the quotient rule.

   $$f(x)=\frac{3x}{x^2+2}$$

   Answer: $\frac{3\left(2-x^2\right)}{x^4+4x^2+4}$

   Step-by-step solution

   1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
   2. $$g^{\prime }(x)=3,h^{\prime }(x)=2x$$
   3. $$f^{\prime }(x)=\frac{3\left(2-x^2\right)}{x^4+4x^2+4}$$
10. 10.

    Differentiate using the quotient rule.

    $$f(x)=\frac{x+2}{x+5}$$

    Answer: $\frac{3}{{\left(x+5\right)}^2}$

    Step-by-step solution

    1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
    2. $$g^{\prime }(x)=1,h^{\prime }(x)=1$$
    3. $$f^{\prime }(x)=\frac{3}{{\left(x+5\right)}^2}$$
11. 11.

    Differentiate using the quotient rule.

    $$f(x)=\frac{x^2-1}{x}$$

    Answer: $1+\frac{1}{x^2}$

    Step-by-step solution

    1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
    2. $$g^{\prime }(x)=2x,h^{\prime }(x)=1$$
    3. $$f^{\prime }(x)=1+\frac{1}{x^2}$$
12. 12.

    Differentiate using the quotient rule.

    $$f(x)=\frac{x-4}{x+1}$$

    Answer: $\frac{5}{{\left(x+1\right)}^2}$

    Step-by-step solution

    1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
    2. $$g^{\prime }(x)=1,h^{\prime }(x)=1$$
    3. $$f^{\prime }(x)=\frac{5}{{\left(x+1\right)}^2}$$
13. 13.

    Differentiate using the quotient rule.

    $$f(x)=\frac{2x+1}{x-3}$$

    Answer: $-\frac{7}{{\left(x-3\right)}^2}$

    Step-by-step solution

    1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
    2. $$g^{\prime }(x)=2,h^{\prime }(x)=1$$
    3. $$f^{\prime }(x)=-\frac{7}{{\left(x-3\right)}^2}$$
14. 14.

    Differentiate using the quotient rule.

    $$f(x)=\frac{x^2+3}{x+2}$$

    Answer: $\frac{-x^2+2x\left(x+2\right)-3}{{\left(x+2\right)}^2}$

    Step-by-step solution

    1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
    2. $$g^{\prime }(x)=2x,h^{\prime }(x)=1$$
    3. $$f^{\prime }(x)=\frac{-x^2+2x\left(x+2\right)-3}{{\left(x+2\right)}^2}$$
15. 15.

    Differentiate using the quotient rule.

    $$f(x)=\frac{x^3}{x+1}$$

    Answer: $\frac{x^2\left(2x+3\right)}{x^2+2x+1}$

    Step-by-step solution

    1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
    2. $$g^{\prime }(x)=3x^2,h^{\prime }(x)=1$$
    3. $$f^{\prime }(x)=\frac{x^2\left(2x+3\right)}{x^2+2x+1}$$
16. 16.

    Differentiate using the quotient rule.

    $$f(x)=\frac{5x}{x-2}$$

    Answer: $-\frac{10}{{\left(x-2\right)}^2}$

    Step-by-step solution

    1. $$\text{Quotient rule: }{\left(\frac{g}{h}\right)}^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}$$
    2. $$g^{\prime }(x)=5,h^{\prime }(x)=1$$
    3. $$f^{\prime }(x)=-\frac{10}{{\left(x-2\right)}^2}$$

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