1. CHAPTER 2 2.4 Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h  0 ( f(a + h) – f(a)) / h if this limit exists. f’(a)= lim x  a ( f(x) – f(a)) / (x – a). Instantaneous rates of change appear so often in applications that they deserve a special name.

2. Example Find the derivative of the function f(x) = 1+ x – 2x 2 at the number a.

3. The tangent line to y = f(x) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a. The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.

4. Example If g(x) = 1 – x 3, find g’(0) and use it to find an equation of the tangent line to the curve y = 1 – x 3 at the point (0,1).

5. Example State f and a in the following limit which represents the derivative of some function f at some number a. lim h -- > 0 ( (2 + h) 3 – 8 ) / h.

6. CHAPTER 2 2.4 Continuity f’(x) = lim h -- > 0 ( f(x + h) – f(x)) / h The Derivative as a Function The derivative at a number a is a number. If we let a vary, x=a, then the derivate will be a function as well.

7. Example Find the derivative of the function f(x) =  1+ x and find the domain of f’. ____

8. Definition A function is differentiable at a if f’(a) exists. It is differentiable on an open interval (a,b) or (a,  ) or (- ,  ) if it is differentiable at every number in the interval. f Theorem If f is differentiable at a, then f is continuous at a. Can a function be continuous but not differentiable at some number?

9. Example Show that f(x) = | x – 6 | is not differentiable at 6. Find the formula for f’ and sketch its graph.

10. The Second Derivative If f is differentiable function, then its derivative f’ is also a function, so f’ may have a derivative of its own, denoted by ( f’)’ = f”. This new function f” is called the second derivative of f.

11. Example Find f” for f(x) = 3x 3 – 4x 2 +7.

12. Various notations for n th derivative of the function y=f(x): y n = f n (x) = (d n y) / (d x n ) =D n x

13. Example If f(x) = x 3 – 2x 2 + 9, find 3 rd and 4 th derivatives of f.

14. Example Based on the following animation, state whether the function is differentiable at x=0. animation