Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Differentiation Techniques: The Power and Sum-Difference Rules OBJECTIVES  Differentiate using the Power Rule or the Sum-Difference Rule.  Differentiate a constant or a constant times a function.  Determine points at which a tangent line has a specified slope. 4.1

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Leibniz’s Notation: When y is a function of x, we will also designate the derivative,, as which is read “the derivative of y with respect to x.” 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 1: The Power Rule For any real number k, 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: Differentiate each of the following: a) b) c) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2: Differentiate: a) b) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 2: The derivative of a constant function is 0. That is, 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 3: The derivative of a constant times a function is the constant times the derivative of the function. That is, 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3: Find each of the following derivatives: a) b) c) a) b) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 (concluded): c) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 4: The Sum-Difference Rule Sum: The derivative of a sum is the sum of the derivatives. Difference: The derivative of a difference is the difference of the derivatives. 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5: Find each of the following derivatives: a) b) a) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 (concluded): b) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6: Find the points on the graph of at which the tangent line is horizontal. Recall that the derivative is the slope of the tangent line, and the slope of a horizontal line is 0. Therefore, we wish to find all the points on the graph of f where the derivative of f equals Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): So, for Setting equal to 0: 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): To find the corresponding y-values for these x-values, substitute back into Thus, the tangent line to the graph of is horizontal at the points (0, 0) and (4, 32). 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (concluded): 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7: Find the x values of the points on the graph of at which the tangent line has slope 6. Here we will employ the same strategy as in Example 6, except that we are now concerned with where the derivative equals 6. Recall that we already found that 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7 (continued): Thus, 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7 (continued): If we were asked to find the corresponding y-values, we would substitute these x-values into 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7 (concluded): 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules