Techniques for Finding Derivatives

1. Techniques for Finding Derivatives Lesson 4.1

2. 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

3. 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

4. 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?

5. 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

6. Decrease the exponent by 1 Multiply the function by the exponent Power Rule • For f(x) = xn • With any real number n • Then

7. Constant Times A Function • What happens when we have a constant times a function? • Example • The rule is • So

8. 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

9. Try It Out • Apply all these rules to take the derivatives of the following functions.

10. 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

11. 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?

12. Assignment • Lesson 4.1A • Page 248 • Exercises 1 – 45 odd • Lesson 4.1B • Page 250 • Exercises 51 – 73 odd