2. Aim: How do we find the derivative by limit process? Do Now: Find the slope of the secant line in terms of x and h. y x (x, f(x)) (x + h, f(x + h)) h f(x + h) – f(x)

3. y x (x, f(x)) (x + h, f(x + h)) h f(x + h) – f(x) As h approaches 0, the secant line approaches the tangent line (x, f(x))

4. Definition of the Slope of a Graph The slope m of the graph of f at the point (x, f(x)) is equal to the slope of its tangent line at (x, f(x)), and is given by provided this limit exists Use the limit process, we can find the exact slope of the tangent line This quotient is called different of quotient

5. Find the slope of the graph f(x) = x 2 at the point (–2, 4) Begin by find an expression that represents the slope of a secant line at the point (–2, 4). Set up different quotient Use f(x) = x 2 Expand terms simplify Factor and cancel simplify The graph has a slope of – 4 at the point (–2, 4).

6. Find the formula for the slope of the graph of f(x) = x 2 + 1 What is the slope at the points (-1,2) and (2,5)? Use the formula m = 2x to find the slope at point (-1,2) the slope is m = 2(-1) = – 2. at the point (2,5), the slope is m = 2(2) = 4

7. Definition of the Derivative The derivative of f at x is given by Remember that the derivative f’(x) is a formula for the slope of the tangent line to the graph of f at the point (x, f(x)).

8. Find the derivative of f(x) = 3x 2 – 2x f’(x) = 6x – 2 Find f’(x) for. Then find the slope of the graph of f at the points (1,1) and (4,2). At the point (1,1), the slope is f’(1) = 1/2. At the point (4,2), the slope is f’(4) = 1/4