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"Calculus and its applications: 1.4" - Differentiation using limits of difference quotients have objective: find derivatives and values of derivatives, find equations of tangent lines. | 2012 Pearson Education, Inc. All rights reserved Slide 1.4- Differentiation Using Limits of Difference Quotients OBJECTIVE Find derivatives and values of derivatives Find equations of tangent lines 2012 Pearson Education, Inc. All rights reserved Slide 1.4- DEFINITION: The slope of the tangent line at (x, f(x)) is This limit is also the instantaneous rate of change of f(x) at x. 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- DEFINITION: For a function y = f (x), its derivative at x is the function defined by provided the limit exists. If exists, then we say that f is differentiable at x. 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- Example 1: For find . Then find and . 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- Example 1 (concluded): 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- Example 2: For find . Then find and . 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- Example 2 (concluded): 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- 1.4 Differentiation Using Limits of Difference Quotients Quick Check 1 Use the results from Examples 1 and 2 to find the derivative and then calculate and . Interpret these results. From Example 1, we know that the derivative of , and from Example 2, we know that the derivative of . Using the Limit Property L3, we then know that . 2012 Pearson Education, Inc. All rights reserved Slide 1.4- 1.4 Differentiation Using Limits of Difference Quotients Quick Check 1 Concluded Now, we plug in into our new derivative formula: Next, we plug in into our new derivative formula: These . | 2012 Pearson Education, Inc. All rights reserved Slide 1.4- Differentiation Using Limits of Difference Quotients OBJECTIVE Find derivatives and values of derivatives Find equations of tangent lines 2012 Pearson Education, Inc. All rights reserved Slide 1.4- DEFINITION: The slope of the tangent line at (x, f(x)) is This limit is also the instantaneous rate of change of f(x) at x. 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- DEFINITION: For a function y = f (x), its derivative at x is the function defined by provided the limit exists. If exists, then we say that f is differentiable at x. 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- Example 1: For find . Then find and . 1.4 Differentiation Using Limits of Difference Quotients 2012 Pearson Education, Inc. All rights reserved Slide 1.4- Example 1 (concluded): 1.4 Differentiation
