Unit 2 Differentiation/Derivatives
Key Unit competence: Solve Economical, Production, and Financial relatedproblems using derivatives.
Introductory activity
Consider functions y = 3x − 2 (1), and y = x2 +1 (2)
a) complete the following table for each of the two functions:
2.1 Differentiation from first principles
2.1.1. Average rate of change of a function
Learning Activity 2.1.1
Suppose that the profit by selling x units of an item is modeled by the
equation, P(x) = 4x2 − 5x + 3, and x assumes values 2 and 5 ,respectively.
Find:
a) The change in x
b) The values of P for x = 2 and x = 5, respectively. Hence, find the
change in P
c) Find the ratio of the change in P to the change in x
d) Give a word with the same meaning as ratio
CONTENT SUMMARY
The average rate of change of function , y = f (x) as the independent variable
Application activity 2.1.1
2.1.2. Instantaneous rate of change of a functionLearning Activity 2.1.2
Consider function y = f (x) = x2 +1, and the changes in x from x0 = 2 to
x1 , where x1 assumes consecutively values x1 = 2.1; x1 = 2.01; x1 = 2.001;...a) Complete the following table:
CONTENT SUMMARY
Application activity 2.1.2
2.2. Rules for differentiation
2.2.1. Differentiation of polynomial functions
Learning Activity 2.2.1
CONTENT SUMMARY
Application activity 2.2.1
2.2.2. Differentiation of product functions
Learning Activity 2.2.2
CONTENT SUMMARY
Application activity 2.2.2
2.2.3. Differentiation of power functions
Learning Activity 2.2.3
CONTENT SUMMARY
Application activity 2.2.3
2.2.4. Differentiation of the composite function (The chain rule)
Learning Activity 2.2.4
CONTENT SUMMARY
Application activity 2.2.4
Use the chain rule to find the derivative of:
a) y = (7x + 8)2
b) y = (4x − 5)3
2.2.5. Differentiation of quotient functions
Learning Activity 2.2.5
CONTENT SUMMARY
Application activity 2.2.4
2.2.6. Differentiation of logarithmic functions
Learning Activity 2.2.6
CONTENT SUMMARY
Application activity 2.2.6
2.2.7. Differentiation of exponential functions
Learning Activity 2.2.7
CONTENT SUMMARY
2.3. Some applications of derivatives in Mathematics
2.3.1. Equation of the tangent to the graph of a function at a point.
Learning Activity 2.3.1
CONTENT SUMMARY
Application activity 2.3.1
2.3.2. Hospital’s rule.
Learning Activity 2.3.2
CONTENT SUMMARY
Application activity 2.3.2
Evaluate the following limits:
End of unit assessment 2
