Download presentation

1
**3.3 Differentiation Rules**

Colorado National Monument Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts Photo by Vickie Kelly, 2003

2
If the derivative of a function is its slope, then for a constant function, the derivative must be zero for all x. example: The derivative of a constant function is zero.

3
We saw that if , This is part of a larger pattern! examples: “power rule”

4
**constant multiple rule:**

examples:

5
**sum and difference rules:**

(Each term is treated separately)

6
Example: Find the horizontal tangents of: Horizontal tangents: (slope of y) = (derivative of y) = zero. Substitute these x values into the original equation to generate coordinate pairs: (0, 2), (-1, 1) and (1,1) … and write tangent lines: (So we expect to see two horizontal tangents, intersecting the curve at three points.)

11
**First derivative y’ (slope) is zero at:**

…where y has slopes of zero The derivative of y is zero at x= -1, 0, 1…

12
product rule: Notice: the derivative of a product is not just the product of the two derivatives. (Our authors write this rule in the other order!) This rule can be “spoken” as: d(u∙v) = v ∙ du + u ∙ dv = (2x3 + 5x) (2x) + (x2 + 3) (6x2 + 5) = (4x4 + 10x2) + (6x4 + 23x2 + 15) = 10x4 + 33x2 + 15 u = x2 + 3 v = 2x3 + 5x du/dx = dv/dx = 2x + 0 2∙3x2 + 5

13
quotient rule: or u = 2x3 + 5x v = x2 + 3 du = dv =

14
**Higher Order Derivatives:**

is the first derivative of y with respect to x. is the second derivative (“y double prime”), the derivative of the first derivative. is the third derivative (“y triple prime”), the derivative of the second derivative. We will see later what these higher-order derivatives might mean in “the real world!” p

Similar presentations

OK

Calculus Warm-Up Find the derivative by the limit process, then find the equation of the line tangent to the graph at x=2.

Calculus Warm-Up Find the derivative by the limit process, then find the equation of the line tangent to the graph at x=2.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on save tigers in india download music Ppt on engineering college life Ppt on condition based maintenance ppt Ppt on plants and trees Ppt on emerging technologies in computer science Ppt on save energy save environment Ppt on motivation for employees Ppt on edge detection in image Ppt on world ending in 2012 Extraocular muscles anatomy and physiology ppt on cells