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Techniques for Finding Derivatives. Lesson 4.1. Limitations of the Definition. Recall our use of the definition of the derivative This worked OK for simple functions Becomes unwieldy for other common functions Higher degree polynomials Trig functions. Other Ways to Represent The Derivative.

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limitations of the definition
Limitations of the Definition
  • Recall our use of the definition of the derivative
  • This worked OK for simple functions
  • Becomes unwieldy for other common functions
    • Higher degree polynomials
    • Trig functions
other ways to represent the derivative
Other Ways to Represent The Derivative
  • Previous chapter
    • For a function f(x), we used f \'(x)
  • To show derivative taken with respect to a variable
    • When y is a function of xshows "derivative of y with respect to x"
  • Other representations
constant rule
Constant Rule
  • Given f(x) = k
    • A constant function
  • Then
  • When we evaluate this we get
  • We conclude when f(x) = k
    • f \'(x) = 0

How does this fit with our understanding that the derivative is the graph of the slope values?

power rule

What pattern do you see?

Power Rule
  • Consider f(x) = x3
  • Use the definition to determine the derivative.
  • Now let h → 0
    • f(x) = x3
    • f \'(x) = 3x2
power rule6

Decrease the exponent by 1

Multiply the function by the exponent

Power Rule
  • For f(x) = xn
    • With any real number n
  • Then
constant times a function
Constant Times A Function
  • What happens when we have a constant times a function?
    • Example
  • The rule is
  • So
sum or difference rule
Sum Or Difference Rule
  • Consider a function which is the sum of two other functions
    • Example :
  • The derivative of f(x) is
  • The derivative of the sum is the sum of the derivatives
try it out
Try It Out
  • Apply all these rules to take the derivatives of the following functions.
marginal analysis
Marginal Analysis
  • Economists use the word "marginal" to refer to rates of change.
  • When we have a function which represents
    • Cost
    • Profit
    • Demand
  • Then the marginal cost (or profit, or demand) is given by the derivative
marginal analysis11
Marginal Analysis
  • When the sales of a product is a function of time t = number of years
  • What is the rate of change or the marginal sales function?
    • What is the rate of change after 3 years?
    • After 10 years?
assignment
Assignment
  • Lesson 4.1A
  • Page 248
  • Exercises 1 – 45 odd
  • Lesson 4.1B
  • Page 250
  • Exercises 51 – 73 odd
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