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 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 x shows "derivative of y with respect to x" Other representations

Constant Rule Given f(x) = k Then When we evaluate this we get 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 Consider f(x) = x3 Use the definition to determine the derivative. Now let h → 0 f(x) = x3 f '(x) = 3x2 What pattern do you see?

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

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

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 Apply all these rules to take the derivatives of the following functions.

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 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 Lesson 4.1A Page 248 Exercises 1 – 45 odd Lesson 4.1B