The portal has been deactivated. Please contact your portal admin.

Lesson Worksheet: The Product Rule Mathematics • Higher Education

In this worksheet, we will practice finding the derivative of a function using the product rule.

Q1:

Consider the functions ๐‘“(๐‘ฅ)=๐‘ฅ, ๐‘”(๐‘ฅ)=๐‘ฅ๏Šจ.

Find ๐‘“(๐‘ฅ)๏Ž˜ and ๐‘”(๐‘ฅ)๏Ž˜.

  • A๐‘“(๐‘ฅ)=1๏Ž˜, ๐‘”(๐‘ฅ)=2๐‘ฅ๏Ž˜
  • B๐‘“(๐‘ฅ)=0๏Ž˜, ๐‘”(๐‘ฅ)=2๐‘ฅ๏Ž˜
  • C๐‘“(๐‘ฅ)=1๏Ž˜, ๐‘”(๐‘ฅ)=2๏Ž˜
  • D๐‘“(๐‘ฅ)=1๏Ž˜, ๐‘”(๐‘ฅ)=2๐‘ฅ๏Ž˜๏Šจ

Find ๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)๏Ž˜๏Ž˜.

  • A๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)=2๏Ž˜๏Ž˜
  • B๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)=0๏Ž˜๏Ž˜
  • C๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)=2๐‘ฅ๏Ž˜๏Ž˜
  • D๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)=2๐‘ฅ๏Ž˜๏Ž˜๏Šจ

Given that ๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)=๐‘ฅ๏Šฉ, find its derivative.

  • A3๐‘ฅ
  • B3๐‘ฅ๏Šฉ
  • C2๐‘ฅ๏Šจ
  • D3๐‘ฅ๏Šจ

Q2:

Find dd๐‘ฆ๐‘ฅ at ๐‘ฅ=โˆ’2 when ๐‘ฆ=(4๐‘ฅโˆ’1)๏€นโˆ’3๐‘ฅ+7๏…๏Šฉ.

Q3:

Complete the following: dd๐‘ฅ๏€บ๐‘ฅโˆš๐‘ฅ๏†=๏Šจ.

  • A๐‘ฅ2โˆš๐‘ฅโˆ’2๐‘ฅโˆš๐‘ฅ๏Šจ
  • B๐‘ฅ2โˆš๐‘ฅ+2๐‘ฅโˆš๐‘ฅ๏Šจ
  • C๐‘ฅโˆš๐‘ฅ+2๐‘ฅโˆš๐‘ฅ๏Šจ
  • D๐‘ฅโˆš๐‘ฅ
  • E2๐‘ฅโˆš๐‘ฅ

Q4:

Find the first derivative of the function ๐‘“(๐‘ฅ)=๏€น2๐‘ฅ+๐‘ฅโˆ’5๏…๏€ผ๐‘ฅ+3โˆš๐‘ฅโˆ’3๐‘ฅ๏ˆ๏Šช๏Šจ.

  • A12๐‘ฅ+27๐‘ฅโˆš๐‘ฅโˆ’15๐‘ฅ+92โˆš๐‘ฅโˆ’10๐‘ฅโˆ’152โˆš๐‘ฅโˆ’15๐‘ฅ๏Šซ๏Šฉ๏Šจ๏Šจ
  • B8๐‘ฅ+27โˆš๐‘ฅโˆ’18๐‘ฅโˆ’15๐‘ฅ+๐‘ฅโˆ’15โˆš๐‘ฅ๏Šซ๏Šฏ๏Šช๏Šฉ๏Šฉ๏Šญ
  • C8๐‘ฅ+27โˆš๐‘ฅโˆ’18๐‘ฅโˆ’15๐‘ฅ+๐‘ฅโˆ’15โˆš๐‘ฅ๏Šซ๏Šญ๏Šช๏Šจ๏Šจ๏Šซ
  • D8๐‘ฅ+27โˆš๐‘ฅโˆ’18๐‘ฅโˆ’15๐‘ฅ+๐‘ฅโˆ’15โˆš๐‘ฅ๏Šช๏Šซ๏Šฉ๏Šฉ

Q5:

Find the first derivative of ๐‘“(๐‘ฅ)=๏€น9๐‘ฅโˆ’๐‘ฅโˆ’7๏…๏€น7๐‘ฅโˆ’8๐‘ฅโˆ’7๏…๏Šจ๏Šจ at ๐‘ฅ=โˆ’1.

Q6:

True or False: If ๐‘“(1)=7, ๐‘“โ€ฒ(1)=3, ๐‘”(1)=10, and ๐‘”โ€ฒ(1)=7, then (๐‘“๐‘”)โ€ฒ(1)=79.

  • ATrue
  • BFalse

Q7:

Suppose that ๐‘“(๐‘ฅ)=(โˆ’2๐‘ฅ+๐‘Ž)๏€น3๐‘ฅโˆ’๐‘Ž๏…๏Šจ and ๐‘“โ€ฒ(โˆ’1)=โˆ’10. Determine ๐‘Ž.

Q8:

Let ๐‘”(๐‘ฅ)=โˆ’3๐‘“(๐‘ฅ)[โ„Ž(๐‘ฅ)โˆ’1]. If ๐‘“โ€ฒ(โˆ’4)=โˆ’1, โ„Žโ€ฒ(โˆ’4)=โˆ’9, โ„Ž(โˆ’4)=โˆ’6, and ๐‘“(โˆ’4)=โˆ’1, find ๐‘”โ€ฒ(โˆ’4).

Q9:

The product rule says that (๐‘“๐‘”)=๐‘“๐‘”+๐‘“๐‘”๏Ž˜๏Ž˜๏Ž˜. Use this to derive a formula for the derivative (๐‘“๐‘”โ„Ž)๏Ž˜.

  • A๐‘“๐‘”โ„Ž+๐‘“๐‘”โ„Ž๏Ž˜๏Ž˜
  • B๐‘“๐‘”โ„Žโˆ’๐‘“๐‘”โ„Žโˆ’๐‘“๐‘”โ„Ž๏Ž˜๏Ž˜๏Ž˜
  • C๐‘“๐‘”โ„Ž+๐‘“๐‘”โ„Ž๏Ž˜๏Ž˜๏Ž˜
  • D๐‘“๐‘”โ„Ž+๐‘“๐‘”โ„Ž+๐‘“๐‘”โ„Ž๏Ž˜๏Ž˜
  • E๐‘“๐‘”โ„Ž+๐‘“๐‘”โ„Ž+๐‘“๐‘”โ„Ž๏Ž˜๏Ž˜๏Ž˜

Q10:

Find the first derivative of ๐‘“(๐‘ฅ)=๏€น๐‘ฅ+4๏…๏€บ3๐‘ฅโˆš๐‘ฅโˆ’7๏†๏€บ3๐‘ฅโˆš๐‘ฅ+7๏†๏Šฎ at ๐‘ฅ=โˆ’1.

This lesson includes 52 additional questions and 198 additional question variations for subscribers.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.