Module 11: THE DERIVATIVE--USING THE RULES
Exploring Calculus

This module will test your proficiency at using the various rules for differentiation which you have recently learned. Upon completing this module, you should be able to differentiate a variety of functions using all of the rules you have learned in your text.

Skill Exercises

Find the derivative of the functions in Qs. 1-10 at the given x-value using the proper rule(s).

Show all of your work !

Check your answers using the graph.

1. y = x² + 3x + 5

at x = 0

2. y = (x³ + 7) (x² + 5x + 4)

at x = -1

3. 

at x = 1

4.

at x = 4

5. y = -3 (x² + 2)⁵

at x = 1

6.

at x = 2

7. y = sin 2x

at

8. y = cos (x² + 4)

at x = -0.5

9.

at

10.

at x = 0

11. Find the equation of the line tangent to the curve

at x = 3 .

12. Find the equation of the line tangent to the curve y = x² - 5x - 6

at x = 2.5 .

13. Given the function, find all of the x-values where the tangent line is horizontal.

    a. How does the sign of the slope of the tangent line change on either side of these x-values?

    b. What is the behavior of the function at each of these x-values?

14. Given any function

    a. When the slope of the tangent line changes sign from positive to negative about some value, the y-value of the function is _____ than all of the surrounding y-values at this x-value.

    b. When the slope of the tangent line changes sign from negative to positive about some x-value, the y-value of the function is
_____ than all of the surrounding  y-values at this x-value.

Exploring Calculus