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

Lesson Worksheet: Rate of Change and Derivatives Mathematics • Higher Education

In this worksheet, we will practice finding the instantaneous rate of change for a function using derivatives and applying this in real-world problems.


Evaluate the instantaneous rate of change of 𝑓(π‘₯)=7π‘₯+9 at π‘₯=π‘₯.

  • A7π‘₯+14π‘₯β„Ž+7β„Ž+9
  • B14π‘₯
  • C7β„Ž+14π‘₯
  • D7β„Ž+14π‘₯β„ŽοŠ¨οŠ§


Evaluate the rate of change of 𝑓(π‘₯)=βˆ’94π‘₯βˆ’7 at π‘₯=4.

  • A4
  • B49
  • Cβˆ’49
  • Dβˆ’1


Evaluate the instantaneous rate of change of 𝑓(π‘₯)=√6π‘₯+7 at π‘₯=3.

  • A3
  • B110
  • C35
  • D65
  • E5


Evaluate the rate of change of 𝑓(π‘₯)=6π‘₯+77π‘₯ at π‘₯=3.

  • A4763
  • B6163
  • C4721
  • D6121


The biomass of a bacterial culture in milligrams as a function of time in minutes is given by 𝑓(𝑑)=71𝑑+63. What is the instantaneous rate of growth of the culture when 𝑑=2minutes?


The distance in meters traveled by a body in 𝑑 seconds is 𝑆=9𝑑+5𝑑+7. What is the instantaneous rate of change of 𝑆 with respect to 𝑑 when 𝑑=11?


A circular disk preserves its shape as it shrinks. What is the rate of change of its area with respect to radius when the radius is 59 cm?

  • A118πœ‹ cm2/cm
  • Bβˆ’59πœ‹ cm2/cm
  • C59πœ‹ cm2/cm
  • Dβˆ’118πœ‹ cm2/cm


Let 𝑓(π‘₯)=π‘Žπ‘₯+π‘οŠ¨ with constants π‘Ž, 𝑏. Determine the constants for which the slope of the tangent to the curve 𝑦=𝑓(π‘₯) at (βˆ’2,29) is βˆ’24.

  • Aπ‘Ž=12, 𝑏=βˆ’19
  • Bπ‘Ž=6, 𝑏=5
  • Cπ‘Ž=βˆ’3, 𝑏=41


Let 𝑓(π‘₯)=5+π‘Žπ‘₯+𝑏π‘₯. Suppose that the change in 𝑓(π‘₯) as π‘₯ goes from βˆ’1 to 2 is 6 and that the rate of change of 𝑓(π‘₯) at π‘₯=2 is 17. Determine π‘Ž and 𝑏.

  • Aπ‘Ž=βˆ’6, 𝑏=10
  • Bπ‘Ž=6, 𝑏=βˆ’2
  • Cπ‘Ž=βˆ’1, 𝑏=3
  • Dπ‘Ž=βˆ’3, 𝑏=5


If the function 𝑓(π‘₯)=5π‘₯+74π‘₯+2, determine its rate of change when π‘₯=2.

  • Aβˆ’950
  • B1710
  • C18289
  • Dβˆ’95
  • E5950

This lesson includes 46 additional questions and 441 additional question variations for subscribers.

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