Presentation is loading. Please wait.

Presentation is loading. Please wait.

2.2 The derivative as a function

Similar presentations


Presentation on theme: "2.2 The derivative as a function"— Presentation transcript:

1 2.2 The derivative as a function
In Section 2.1 we considered the derivative at a fixed number a. Now let number a vary. If we replace number a by a variable x, then the derivative can be interpreted as a function of x : Alternative notations for the derivative: D and d / dx are called differentiation operators. dy / dx should not be regarded as a ratio.

2 The derivative is the slope of the original function.

3

4 Differentiable functions
A function f is differentiable at a if f ′(a) exists. It is differentiable on an open interval (a,b) [ or (a,) or (- , a) or (- , ) ] if it is differentiable at every number in the interval. Theorem: If f is differentiable at a, then f is continuous at a. Note: The converse is false: there are functions that are continuous but not differentiable. Example: f(x) = | x |

5 To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at: corner cusp discontinuity vertical tangent

6 Higher Order Derivatives:
is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative. We will learn later what these higher order derivatives are used for. is the fourth derivative.


Download ppt "2.2 The derivative as a function"

Similar presentations


Ads by Google