Product 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 product rule.

   $$f(x)=x^2\sin \left(x\right)$$

   Answer: $x\left(x\cos \left(x\right)+2\sin \left(x\right)\right)$

   Step-by-step solution

   1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
   2. $$g^{\prime }(x)=2x,h^{\prime }(x)=\cos \left(x\right)$$
   3. $$f^{\prime }(x)=x\left(x\cos \left(x\right)+2\sin \left(x\right)\right)$$
2. 2.

   Differentiate using the product rule.

   $$f(x)=x{e}^x$$

   Answer: $\left(x+1\right)e^x$

   Step-by-step solution

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

   Differentiate using the product rule.

   $$f(x)=\left(x+1\right)\cos \left(x\right)$$

   Answer: $-\left(x+1\right)\sin \left(x\right)+\cos \left(x\right)$

   Step-by-step solution

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

   Differentiate using the product rule.

   $$f(x)=x^3{e}^x$$

   Answer: $x^2\left(x+3\right)e^x$

   Step-by-step solution

   1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
   2. $$g^{\prime }(x)=3x^2,h^{\prime }(x)=e^x$$
   3. $$f^{\prime }(x)=x^2\left(x+3\right)e^x$$
5. 5.

   Differentiate using the product rule.

   $$f(x)=2x\sin \left(x\right)$$

   Answer: $2x\cos \left(x\right)+2\sin \left(x\right)$

   Step-by-step solution

   1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
   2. $$g^{\prime }(x)=2,h^{\prime }(x)=\cos \left(x\right)$$
   3. $$f^{\prime }(x)=2x\cos \left(x\right)+2\sin \left(x\right)$$
6. 6.

   Differentiate using the product rule.

   $$f(x)=\left(x^2+1\right)\cos \left(x\right)$$

   Answer: $2x\cos \left(x\right)-\left(x^2+1\right)\sin \left(x\right)$

   Step-by-step solution

   1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
   2. $$g^{\prime }(x)=2x,h^{\prime }(x)=-\sin \left(x\right)$$
   3. $$f^{\prime }(x)=2x\cos \left(x\right)-\left(x^2+1\right)\sin \left(x\right)$$
7. 7.

   Differentiate using the product rule.

   $$f(x)=x^2\cos \left(x\right)$$

   Answer: $x\left(-x\sin \left(x\right)+2\cos \left(x\right)\right)$

   Step-by-step solution

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

   Differentiate using the product rule.

   $$f(x)=3x{e}^x$$

   Answer: $3\left(x+1\right)e^x$

   Step-by-step solution

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

   Differentiate using the product rule.

   $$f(x)=\left(x-2\right)\sin \left(x\right)$$

   Answer: $\left(x-2\right)\cos \left(x\right)+\sin \left(x\right)$

   Step-by-step solution

   1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
   2. $$g^{\prime }(x)=1,h^{\prime }(x)=\cos \left(x\right)$$
   3. $$f^{\prime }(x)=\left(x-2\right)\cos \left(x\right)+\sin \left(x\right)$$
10. 10.

    Differentiate using the product rule.

    $$f(x)=x^3\sin \left(x\right)$$

    Answer: $x^2\left(x\cos \left(x\right)+3\sin \left(x\right)\right)$

    Step-by-step solution

    1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
    2. $$g^{\prime }(x)=3x^2,h^{\prime }(x)=\cos \left(x\right)$$
    3. $$f^{\prime }(x)=x^2\left(x\cos \left(x\right)+3\sin \left(x\right)\right)$$
11. 11.

    Differentiate using the product rule.

    $$f(x)=\left(x^2+x\right)e^x$$

    Answer: $\left(x^2+3x+1\right)e^x$

    Step-by-step solution

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

    Differentiate using the product rule.

    $$f(x)=5x\cos \left(x\right)$$

    Answer: $-5x\sin \left(x\right)+5\cos \left(x\right)$

    Step-by-step solution

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

    Differentiate using the product rule.

    $$f(x)=\left(x+4\right)e^x$$

    Answer: $\left(x+5\right)e^x$

    Step-by-step solution

    1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
    2. $$g^{\prime }(x)=1,h^{\prime }(x)=e^x$$
    3. $$f^{\prime }(x)=\left(x+5\right)e^x$$
14. 14.

    Differentiate using the product rule.

    $$f(x)=x^4\cos \left(x\right)$$

    Answer: $x^3\left(-x\sin \left(x\right)+4\cos \left(x\right)\right)$

    Step-by-step solution

    1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
    2. $$g^{\prime }(x)=4x^3,h^{\prime }(x)=-\sin \left(x\right)$$
    3. $$f^{\prime }(x)=x^3\left(-x\sin \left(x\right)+4\cos \left(x\right)\right)$$
15. 15.

    Differentiate using the product rule.

    $$f(x)=2x^2\sin \left(x\right)$$

    Answer: $2x\left(x\cos \left(x\right)+2\sin \left(x\right)\right)$

    Step-by-step solution

    1. $$\text{Product rule: }(gh{)}^{\prime }=g^{\prime }h+g{h}^{\prime }$$
    2. $$g^{\prime }(x)=4x,h^{\prime }(x)=\cos \left(x\right)$$
    3. $$f^{\prime }(x)=2x\left(x\cos \left(x\right)+2\sin \left(x\right)\right)$$
16. 16.

    Differentiate using the product rule.

    $$f(x)=\left(x^2-3\right)e^x$$

    Answer: $\left(x^2+2x-3\right)e^x$

    Step-by-step solution

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

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