Definition of the Derivative Using Average Rate (Page 129 - 133 and 160 in the book)

1. f(a+h) – f(a) h Definition of the Derivative Using Average Rate(Page 129 - 133 and 160 in the book) f(a+h) Slope of the line = Average Rate of Change = f(a+h) – f(a) f(a) h h a a+h

2. h h h h a a+h a a+h a a+h a Now, Watch what happens when:Point a is fixed and the size of the interval h shrinks

3. Slope of the line = Average Rate of Change = f(a+h) – f(a) h Slope of the Tangent line = f(a+h) – f(a) h f(a+h) f(a) h a a+h As h shrinks and approaches zero (but not = 0), the line becomes a Tangent Line As h approaches zero

4. Slope of the Tangent line lim f ' (a) = h0 As h approaches zero, or: f(a+h) – f(a) h f(a+h) – f(a) h = h 0 = lim f(a+h) – f(a) h f(a) The slope of the Tangent Line at a is the Derivative, f ' (a) a lim: Limit, as h approaches zero

5. Example: Use the definition of the derivative to obtain the following result: If f(x) = -2x + 3, then f' (x) = -2 lim lim lim lim f' (x) = h0 h0 h0 h0 f(x+h) – f(x) h f (x + h) – f (x) h f' (x) = (-2x - 2h + 3) – (-2x + 3)h = (-2h)h = Solution: Using the definition f (x + h) = -2(x + h) + 3 = (-2x - 2h + 3) = -2

6. Example: Use the definition of the derivative to obtain the following result: If f(x) = x2 - 8x + 9, then f' (x) = 2x - 8 lim lim lim lim lim lim f' (x) = h0 h0 h0 h0 h0 h0 f(x+h) – f(x) h f (x + h) – f (x) h f' (x) = (x2 + 2xh + h2 - 8x - 8h + 9) – ( x2 - 8x + 9)h = (2xh + h2 - 8h)h h (2x + h - 8)h = = (2x + h - 8) = Solution: Using the definition f (x + h) = (x + h)2 - 8(x + h) + 9 = (x2 + 2xh + h2 - 8x -8h + 9) = 2x - 8