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2.3 The Derivative

T. x. 2.3 The Derivative. General idea: introduce the derivative from geometric configuration as the slope of the line tangent to a curve at a given point .

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2.3 The Derivative

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  1. T x 2.3 The Derivative General idea: introduce the derivative from geometric configuration as the slope of the line tangent to a curve at a given point. Method: we are going to find the tangent line as a limiting position for a secant line as points of intersection approach each other. 1

  2. Formalization: 1. Introduce coordinates for the point P: , where 2. Introduce differences between the coordinates of the point Q and point P:Dx and Dy. Now the coordinates of Q are where 3. The slope of the secant line PQ can be written as 2

  3. Formalization (contd): 4. Combining 1 and 2 we can write Dy as 5. The statement QP can now be formalized asDx0, and 3

  4. The formula obtained for the slope of the tangent line constitutes the definition of the derivative. We denote it as This is a derivative taken at one particular point but we can generalize it on any point in the domain of the function: Definition of the Derivative: If the limit exists at some value of x, then the function is called differentiable at this point. If it exists at every point in an interval, the function is differentiable in the whole interval. Other notations: 4

  5. 2.4 Differentiation by the Four-Step Process: Step 1: In the function y=f(x), replace x by x+Dx and y by y+Dy: Step2: Subtract y=f(x) from both sides: Step 3: Divide both sides of the resulting expression by Dx: Step 4: Obtain f’(x) by evaluating 5

  6. Example: Find the derivative of Solution: Step 1. for gives Step 2. gives Step 3. Dividing by Dx, we get + simplify the expression to get rid of Dx in the denominator…. Step 4. 6

  7. Example: Evaluate the derivative of at the points x=1, x=2 Solution: Step 1. Step 2. simplify the expression to get Dx as a factor….. Step 3. Step 4. 7

  8. Example: Find the derivative of Solution: Step 1. Step 2. simplify the expression to get Dx as a factor….. Step 3. Step 4. 8

  9. Homework Section 2.4: 3,5,9,11,15,19,21,23. 9

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