Differentiation Rules. Oct
|
|
|
- Kelly Hawkins
- 5 years ago
- Views:
Transcription
1 Differentiation Rules Oct
2 Differentiability versus Continuity Theorem If f (a) exists, then f is continuous at a. A function whose erivative exists at every point of an interval is continuous an smooth, i.e. it has no sharp corners.
3
4 Builing the Toolbox Theorem (Scalars) If y = f (x) is a ifferentiable function then, (c f (x)) = c f (x) x (c f (x)) = c x f (x) where c is a constant.
5
6 Builing the Toolbox Theorem (Scalars) If y = f (x) is a ifferentiable function then, (c f (x)) = c f (x) x (c f (x)) = c x f (x) where c is a constant. Example Since x x 2 =2x, wehave x 5x 2 =5 2x = 10x.
7 Builing the Toolbox Theorem (Sums) If f an g are ifferentiable functions, then x (f (x)+g(x)) = f (x)+g (x) In wors, the erivative of a sum is the sum of the erivatives.
8
9 Builing the Toolbox Theorem (Sums) If f an g are ifferentiable functions, then x (f (x)+g(x)) = f (x)+g (x) In wors, the erivative of a sum is the sum of the erivatives. x (3x 2 +2x + 7) =
10 Builing the Toolbox Theorem (Sums) If f an g are ifferentiable functions, then x (f (x)+g(x)) = f (x)+g (x) In wors, the erivative of a sum is the sum of the erivatives. x (3x 2 +2x + 7) = x (3x 2 )+ x (2x)+ x (7) =3 2x = 6x+2
11 Calculate x (5x + 3)(x + 2). Caution! The answer is not 5. Expan first!
12 Calculate x (5x + 3)(x + 2). Caution! The answer is not 5. Expan first! (5x + 3)(x + 2) = 5x x +6 x x =5 2x + 13 = 10x+13
13 Builing the Toolbox Theorem (Proucts) If f (x) an g(x) are ifferentiable functions, then x (f (x) g(x)) = f (x) g (x)+g(x) f (x). In wors, the erivative of a prouct is the first times the erivative of the secon, plus the secon times the erivative of the first.
14 Builing the Toolbox Theorem (Proucts) If f (x) an g(x) are ifferentiable functions, then x (f (x) g(x)) = f (x) g (x)+g(x) f (x). In wors, the erivative of a prouct is the first times the erivative of the secon, plus the secon times the erivative of the first. Try 2: Calculate x (5x + 3)(x + 2) : x (5x + 3)(x + 2) =(5x + 3) 1+(x + 2) 5 = 10x+13 f g f g + g f
15
16 Builing the Toolbox Theorem (Reciprocals) If y = f (x) is a ifferentiable function, then wherever f (x) = 0, 1 = f (x) x f (x) f (x) 2.
17
18 Builing the Toolbox Theorem (Reciprocals) If y = f (x) is a ifferentiable function, then wherever f (x) = 0, 1 = f (x) x f (x) f (x) 2. Example: Calculate x 1 x 2.
19 Builing the Toolbox Theorem (Reciprocals) If y = f (x) is a ifferentiable function, then wherever f (x) = 0, 1 = f (x) x f (x) f (x) 2. Example: Here, f (x) =x 2,so 1 x x 2 Calculate x 1 x 2. = 2x (x 2 ) 2 = 2 x 3 = 2x 3
20 Some more examples of reciprocals Rule: 1 x f (x) = f (x). f (x) 2
21 Some more examples of reciprocals Rule: 1 x f (x) = f (x). f (x) 2 1 x x 2 +3x +1 2x +3 = (x 2 +3x +1) 2
22 Some more examples of reciprocals Rule: 1 x f (x) = f (x). f (x) 2 1 x x 2 +3x +1 2x +3 = (x 2 +3x +1) 2 Quotients: x 2 1 x x 2 = (x 2 1 1) +3x +1 x x 2 +3x +1
23 Some more examples of reciprocals Rule: 1 x f (x) = f (x). f (x) 2 1 x x 2 +3x +1 2x +3 = (x 2 +3x +1) 2 Quotients: x 2 1 x x 2 = (x 2 1 1) +3x +1 x x 2 +3x +1 =(x 2 (2x + 3) 1) (x 2 +3x + 1) x 2 +3x +1 (2x)
24 In general, f = x g x 1 f g =
25 In general, f = 1 x g x g 1 = g f x (f )+f x 1 g (prouct rule)
26 In general, f = 1 f x g x g 1 = g x (f )+f x = f g + f g g 2 1 g (prouct rule) (reciprocal rule)
27 In general, f = 1 f x g x g 1 = g x (f )+f x = f g + f g g 2 = f g f g g 2 1 g (prouct rule) (reciprocal rule)
28 In general, f = 1 f x g x g 1 = g x (f )+f x = f g + f g g 2 = f g f g g 2 = g f f g g 2 1 g (prouct rule) (reciprocal rule) (common enominator)
29 In general, f = 1 f x g x g 1 = g x (f )+f x = f g + f g g 2 = f g f g g 2 = g f f g g 2 1 g (prouct rule) (reciprocal rule) (common enominator) Theorem (Quotient rule) Wherever g(x) = 0, x f (x) g(x) = g(x) f (x) f (x) g (x) g(x) 2
30 Builing the Toolbox Theorem Let (f g)(x) =f (g(x)). Assume that g is ifferentiable at the point x an that f is ifferentiable at the point g(x). Then the composite function f gisifferentiable at the point x, an x ((f g)(x)) = f (g(x)) = f (g(x)) g (x). x Using Leibniz s notation: y x = y u u x.
31 Chain rule: Example x Calculate x f (g(x)) = f (g(x)) g (x). x
32 Chain rule: Example x Calculate x f (g(x)) = f (g(x)) g (x). x Here, f (x) = x = x 1/2 an g(x) =x 7 +5.
33 Chain rule: Example x Calculate x f (g(x)) = f (g(x)) g (x). x Here, f (x) = x = x 1/2 an g(x) =x So f (x) = 1 2 x 1/2 = 1 2 x an g (x) =7x 6
34 Chain rule: Example x Calculate x f (g(x)) = f (g(x)) g (x). x Here, f (x) = x = x 1/2 an g(x) =x So an so f (x) = 1 2 x 1/2 = 1 2 x an g (x) =7x 6 x x 7 +5 = 1 2 x x 6.
35 Back to reciprocals: Chain rule: Calculate x f (g(x)) = f (g(x)) g (x). x 1 g(x) = x (g(x)) 1
36 Back to reciprocals: Chain rule: Calculate x Outsie function: f (x) =x 1 Insie function: g(x) f (g(x)) = f (g(x)) g (x). x 1 g(x) = x (g(x)) 1
37 Back to reciprocals: Chain rule: Calculate x Outsie function: f (x) =x 1 Insie function: g(x) f (g(x)) = f (g(x)) g (x). x 1 g(x) = x (g(x)) 1 So since x (x 1 )= x 2 = 1 x 2 an x (g(x)) = g (x),
38 Back to reciprocals: Chain rule: Calculate x Outsie function: f (x) =x 1 Insie function: g(x) f (g(x)) = f (g(x)) g (x). x 1 g(x) = x (g(x)) 1 So since x (x 1 )= x 2 = 1 x 2 an x (g(x)) = g (x), we get 1 = 1 x g(x) (g(x)) 2 g (x) = g (x) (g(x)) 2 just like before!
39
40
41
42
43
44
45
Section 3.1/3.2: Rules of Differentiation
: Rules of Differentiation Math 115 4 February 2018 Overview 1 2 Four Theorem for Derivatives Suppose c is a constant an f, g are ifferentiable functions. Then 1 2 3 4 x (c) = 0 x (x n ) = nx n 1 x [cf
Chapter 3 Definitions and Theorems
Chapter 3 Definitions an Theorems (from 3.1) Definition of Tangent Line with slope of m If f is efine on an open interval containing c an the limit Δy lim Δx 0 Δx = lim f (c + Δx) f (c) = m Δx 0 Δx exists,
2.5 The Chain Rule Brian E. Veitch
2.5 The Chain Rule This is our last ifferentiation rule for this course. It s also one of the most use. The best way to memorize this (along with the other rules) is just by practicing until you can o
THEOREM: THE CONSTANT RULE
MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1:
The derivative of a function f is a new function defined by. f f (x + h) f (x)
Derivatives Definition Te erivative of a function f is a new function efine by f f (x + ) f (x) (x). 0 We will say tat a function f is ifferentiable over an interval (a, b) if if te erivative function
Define each term or concept.
Chapter Differentiation Course Number Section.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
February 21 Math 1190 sec. 63 Spring 2017
February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative
Differentiation Rules c 2002 Donald Kreider and Dwight Lahr
Dierentiation Rules c 00 Donal Kreier an Dwigt Lar Te Power Rule is an example o a ierentiation rule. For unctions o te orm x r, were r is a constant real number, we can simply write own te erivative rater
Chapter 3 Notes, Applied Calculus, Tan
Contents 3.1 Basic Rules of Differentiation.............................. 2 3.2 The Prouct an Quotient Rules............................ 6 3.3 The Chain Rule...................................... 9 3.4
Implicit Differentiation
Implicit Differentiation Implicit Differentiation Using the Chain Rule In the previous section we focuse on the erivatives of composites an saw that THEOREM 20 (Chain Rule) Suppose that u = g(x) is ifferentiable
1 Lecture 18: The chain rule
1 Lecture 18: The chain rule 1.1 Outline Comparing the graphs of sin(x) an sin(2x). The chain rule. The erivative of a x. Some examples. 1.2 Comparing the graphs of sin(x) an sin(2x) We graph f(x) = sin(x)
Section The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions
Section 3.4-3.6 The Chain Rule an Implicit Differentiation with Application on Derivative of Logarithm Functions Ruipeng Shen September 3r, 5th Ruipeng Shen MATH 1ZA3 September 3r, 5th 1 / 3 The Chain
The Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier.
The Chain Rule In the previous section we ha to use a trig ientity to etermine the erivative of. h(x) = sin(2x). We can view h(x) as the composition of two functions. Let g(x) = 2x an f (x) = sin x. Then
Antiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut
Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Definition (Inefinite
x 2 2x 8 (x 4)(x + 2)
Problems With Notation Mathematical notation is very precise. This contrasts with both oral communication an some written English. Correct mathematical notation: x 2 2x 8 (x 4)(x + 2) lim x 4 = lim x 4
Derivatives and the Product Rule
Derivatives an the Prouct Rule James K. Peterson Department of Biological Sciences an Department of Mathematical Sciences Clemson University January 28, 2014 Outline Differentiability Simple Derivatives
0.1 Differentiation Rules
0.1 Differentiation Rules From our previous work we ve seen tat it can be quite a task to calculate te erivative of an arbitrary function. Just working wit a secon-orer polynomial tings get pretty complicate
By writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2)
3.5 Chain Rule 149 3.5 Chain Rule Introuction As iscusse in Section 3.2, the Power Rule is vali for all real number exponents n. In this section we see that a similar rule hols for the erivative of a power
Math Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis
Math 231 - Chapter 2 Essentials of Calculus by James Stewart Prepare by Jason Gais Chapter 2 - Derivatives 21 - Derivatives an Rates of Change Definition A tangent to a curve is a line that intersects
DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS
DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS THE DERIVATIVE AS A FUNCTION f x = lim h 0 f x + h f(x) h Last class we examine the limit of the ifference quotient at a specific x as h 0,
Mathematics 1210 PRACTICE EXAM II Fall 2018 ANSWER KEY
Mathematics 1210 PRACTICE EXAM II Fall 2018 ANSWER KEY 1. Calculate the following: a. 2 x, x(t) = A sin(ωt φ) t2 Solution: Using the chain rule, we have x (t) = A cos(ωt φ)ω = ωa cos(ωt φ) x (t) = ω 2
1. Find the equation of a line passing through point (5, -2) with slope ¾. (State your answer in slope-int. form)
INTRO TO CALCULUS REVIEW FINAL EXAM NAME: DATE: A. Equations of Lines (Review Chapter) y = m + b (Slope-Intercept Form) A + By = C (Stanar Form) y y = m( ) (Point-Slope Form). Fin the equation of a line
Section 7.1: Integration by Parts
Section 7.1: Integration by Parts 1. Introuction to Integration Techniques Unlike ifferentiation where there are a large number of rules which allow you (in principle) to ifferentiate any function, the
13.1: Vector-Valued Functions and Motion in Space, 14.1: Functions of Several Variables, and 14.2: Limits and Continuity in Higher Dimensions
13.1: Vector-Value Functions an Motion in Space, 14.1: Functions of Several Variables, an 14.2: Limits an Continuity in Higher Dimensions TA: Sam Fleischer November 3 Section 13.1: Vector-Value Functions
Rules of Differentiation. Lecture 12. Product and Quotient Rules.
Rules of Differentiation. Lecture 12. Prouct an Quotient Rules. We warne earlier that we can not calculate the erivative of a prouct as the prouct of the erivatives. It is easy to see that this is so.
Topic 2.3: The Geometry of Derivatives of Vector Functions
BSU Math 275 Notes Topic 2.3: The Geometry of Derivatives of Vector Functions Textbook Sections: 13.2 From the Toolbox (what you nee from previous classes): Be able to compute erivatives scalar-value functions
Differentiation Rules and Formulas
Differentiation Rules an Formulas Professor D. Olles December 1, 01 1 Te Definition of te Derivative Consier a function y = f(x) tat is continuous on te interval a, b]. Ten, te slope of te secant line
SYDE 112, LECTURE 1: Review & Antidifferentiation
SYDE 112, LECTURE 1: Review & Antiifferentiation 1 Course Information For a etaile breakown of the course content an available resources, see the Course Outline. Other relevant information for this section
( 3x +1) 2 does not fit the requirement of the power rule that the base be x
Section 3 4A: The Chain Rule Introuction The Power Rule is state as an x raise to a real number If y = x n where n is a real number then y = n x n-1 What if we wante to fin the erivative of a variable
Rules of Differentiation
LECTURE 2 Rules of Differentiation At te en of Capter 2, we finally arrive at te following efinition of te erivative of a function f f x + f x x := x 0 oing so only after an extene iscussion as wat te
Chapter 2 Derivatives
Chapter Derivatives Section. An Intuitive Introuction to Derivatives Consier a function: Slope function: Derivative, f ' For each, the slope of f is the height of f ' Where f has a horizontal tangent line,
Using the definition of the derivative of a function is quite tedious. f (x + h) f (x)
Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit
Solutions to Practice Problems Tuesday, October 28, 2008
Solutions to Practice Problems Tuesay, October 28, 2008 1. The graph of the function f is shown below. Figure 1: The graph of f(x) What is x 1 + f(x)? What is x 1 f(x)? An oes x 1 f(x) exist? If so, what
1 Lecture 20: Implicit differentiation
Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation
Review of Differentiation and Integration for Ordinary Differential Equations
Schreyer Fall 208 Review of Differentiation an Integration for Orinary Differential Equations In this course you will be expecte to be able to ifferentiate an integrate quickly an accurately. Many stuents
SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 1 8 3
SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 8 3 L P f Q L segments L an L 2 to be tangent to the parabola at the transition points P an Q. (See the figure.) To simplify the equations you ecie to place the
Section 2.1 The Derivative and the Tangent Line Problem
Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
Section 3.3 Product and Quotient rules 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.3 Product and Quotient rules 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Differentiation Rules 1 / 12 Motivation Goal: We want to derive rules to find the derivative
Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides
Reference 1: Transformations of Graphs an En Behavior of Polynomial Graphs Transformations of graphs aitive constant constant on the outsie g(x) = + c Make graph of g by aing c to the y-values on the graph
Calculus I Announcements
Slie 1 Calculus I Announcements Office Hours: Amos Eaton 309, Monays 12:50-2:50 Exam 2 is Thursay, October 22n. The stuy guie is now on the course web page. Start stuying now, an make a plan to succee.
Math 115 Section 018 Course Note
Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of
016A Homework 10 Solution
016A Homework 10 Solution Jae-young Park November 2, 2008 4.1 #14 Write each expression in the form of 2 kx or 3 kx, for a suitable constant k; (3 x 3 x/5 ) 5, (16 1/4 16 3/4 ) 3x Solution (3 x 3 x/5 )
1 Definition of the derivative
Math 20A - Calculus by Jon Rogawski Chapter 3 - Differentiation Prepare by Jason Gais Definition of the erivative Remark.. Recall our iscussion of tangent lines from way back. We now rephrase this in terms
MATH 205 Practice Final Exam Name:
MATH 205 Practice Final Eam Name:. (2 points) Consier the function g() = e. (a) (5 points) Ientify the zeroes, vertical asymptotes, an long-term behavior on both sies of this function. Clearly label which
Final Exam Study Guide and Practice Problems Solutions
Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making
Math 210 Midterm #1 Review
Math 20 Miterm # Review This ocument is intene to be a rough outline of what you are expecte to have learne an retaine from this course to be prepare for the first miterm. : Functions Definition: A function
MATH2231-Differentiation (2)
-Differentiation () The Beginnings of Calculus The prime occasion from which arose my iscovery of the metho of the Characteristic Triangle, an other things of the same sort, happene at a time when I ha
102 Problems Calculus AB Students Should Know: Solutions. 18. product rule d. 19. d sin x. 20. chain rule d e 3x2) = e 3x2 ( 6x) = 6xe 3x2
Problems Calculus AB Stuents Shoul Know: Solutions. + ) = + =. chain rule ) e = e = e. ) =. ) = ln.. + + ) = + = = +. ln ) =. ) log ) =. sin ) = cos. cos ) = sin. tan ) = sec. cot ) = csc. sec ) = sec
Lecture 14 September 26, Today. WH 3 now posted Due Tues. Oct. 2, 2018 Quiz 4 tomorrow. Differentiation summary Related rates
Lecture 4 September 6, 08 WH 3 now poste Due Tues. Oct., 08 Quiz 4 tomorrow Toay Differentiation summary Relate rates Differentiation Summary Basic erivatives (memorize) x x x c = 0 sin x = cos x cos x
Calculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10
Calculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10 This is a set of practice test problems for Chapter 4. This is in no way an inclusive set of problems there can be other types of problems
MAT01A1: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules
MAT01A1: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 22 Marc 2017 Semester Test 1 Scripts will be available for collection from Tursay morning. For marking
1 Lecture 13: The derivative as a function.
1 Lecture 13: Te erivative as a function. 1.1 Outline Definition of te erivative as a function. efinitions of ifferentiability. Power rule, erivative te exponential function Derivative of a sum an a multiple
The Natural Logarithm
The Natural Logarithm -28-208 In earlier courses, you may have seen logarithms efine in terms of raising bases to powers. For eample, log 2 8 = 3 because 2 3 = 8. In those terms, the natural logarithm
2.1 Derivatives and Rates of Change
1a 1b 2.1 Derivatives an Rates of Change Tangent Lines Example. Consier y f x x 2 0 2 x-, 0 4 y-, f(x) axes, curve C Consier a smooth curve C. A line tangent to C at a point P both intersects C at P an
Math 1A Midterm 2 Fall 2015 Riverside City College (Use this as a Review)
Name Date Miterm Score Overall Grae Math A Miterm 2 Fall 205 Riversie City College (Use this as a Review) Instructions: All work is to be shown, legible, simplifie an answers are to be boxe in the space
MAT137 Calculus! Lecture 6
MAT137 Calculus! Lecture 6 Today: 3.2 Differentiation Rules; 3.3 Derivatives of higher order. 3.4 Related rates 3.5 Chain Rule 3.6 Derivative of Trig. Functions Next: 3.7 Implicit Differentiation 4.10
d dx [xn ] = nx n 1. (1) dy dx = 4x4 1 = 4x 3. Theorem 1.3 (Derivative of a constant function). If f(x) = k and k is a constant, then f (x) = 0.
![d dx [xn ] = nx n 1. (1) dy dx = 4x4 1 = 4x 3. Theorem 1.3 (Derivative of a constant function). If f(x) = k and k is a constant, then f (x) = 0.](/thumbs/77/76592604.jpg "d dx [xn ] = nx n 1. (1) dy dx = 4x4 1 = 4x 3. Theorem 1.3 (Derivative of a constant function). If f(x) = k and k is a constant, then f (x) = 0.")
Calculus refresher Disclaimer: I claim no original content on this ocument, which is mostly a summary-rewrite of what any stanar college calculus book offers. (Here I ve use Calculus by Dennis Zill.) I
MAT1A01: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules
MAT1A01: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 22 Marc 2016 Semester Test 1 Results ave been publise on Blackboar uner My Graes. Scripts will be available
MATH 1300 Lecture Notes Wednesday, September 25, 2013
MATH 300 Lecture Notes Wenesay, September 25, 203. Section 3. of HH - Powers an Polynomials In this section 3., you are given several ifferentiation rules that, taken altogether, allow you to quickly an
by using the derivative rules. o Building blocks: d
Calculus for Business an Social Sciences - Prof D Yuen Eam Review version /9/01 Check website for any poste typos an upates Eam is on Sections, 5, 6,, 1,, Derivatives Rules Know how to fin the formula
Math 190 Chapter 3 Lecture Notes. Professor Miguel Ornelas
Math 190 Chapter 3 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 190 Lecture Notes Section 3.1 Section 3.1 Derivatives of Polynomials an Exponential Functions Derivative of a Constant Function
Integration via a Change of Variables
LECTURE 33 Integration via a Change of Variables In Lectures an 3 where we evelope a general technique for computing erivatives that was base on two ifferent kins of results. First, we ha a table that
Integration: Using the chain rule in reverse
Mathematics Learning Centre Integration: Using the chain rule in reverse Mary Barnes c 999 University of Syney Mathematics Learning Centre, University of Syney Using the Chain Rule in Reverse Recall that
Inverse Functions. Review from Last Time: The Derivative of y = ln x. [ln. Last time we saw that
Inverse Functions Review from Last Time: The Derivative of y = ln Last time we saw that THEOREM 22.0.. The natural log function is ifferentiable an More generally, the chain rule version is ln ) =. ln
Differentiation ( , 9.5)
Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the
Chapter 1. Functions, Graphs, and Limits
Review for Final Exam Lecturer: Sangwook Kim Office : Science & Tech I, 226D math.gmu.eu/ skim22 Chapter 1. Functions, Graphs, an Limits A function is a rule that assigns to each objects in a set A exactly
Computing Derivatives J. Douglas Child, Ph.D. Rollins College Winter Park, FL
Computing Derivatives by J. Douglas Chil, Ph.D. Rollins College Winter Park, FL ii Computing Inefinite Integrals Important notice regaring book materials Texas Instruments makes no warranty, either express
f(x) f(a) Limit definition of the at a point in slope notation.
Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126-235 Definition. Limit Definition of Derivatives at a point
Differentiation Rules Derivatives of Polynomials and Exponential Functions
Derivatives of Polynomials an Exponential Functions Differentiation Rules Derivatives of Polynomials an Exponential Functions Let s start with the simplest of all functions, the constant function f(x)
3.3 The Chain Rule CHAPTER 3. RULES FOR DERIVATIVES 63
CHAPTER 3. RULES FOR DERIVATIVES 63 3.3 The Chain Rule Comments. The Chain Rule is the most interesting one we have learne so far, an it is one of the most complicate we will learn. It is one of the rules
Connecting Algebra to Calculus Indefinite Integrals
Connecting Algebra to Calculus Inefinite Integrals Objective: Fin Antierivatives an use basic integral formulas to fin Inefinite Integrals an make connections to Algebra an Algebra. Stanars: Algebra.0,
The Explicit Form of a Function
Section 3 5 Implicit Differentiation The Eplicit Form of a Function The normal way we see function notation has f () on one sie of an equation an an epression in terms of on the other sie. We know the
Outline. Calculus for the Life Sciences. Average Height and Weight of Girls 1. Chain Rule. Lecture Notes Chain Rule
Outline Calculus for the Life Sciences Lecture Notes Joseph M. Mahaffy, jmahaffy@mail.ssu.eu Department of Mathematics an Statistics Dynamical Systems Group Computational Sciences Research Center San Diego
11.7. Implicit Differentiation. Introduction. Prerequisites. Learning Outcomes
Implicit Differentiation 11.7 Introuction This Section introuces implicit ifferentiation which is use to ifferentiate functions expresse in implicit form (where the variables are foun together). Examples
f (x) f (a) f (a) = lim x a f (a) x a
Differentiability Revisited Recall that the function f is differentiable at a if exists and is finite. f (a) = lim x a f (x) f (a) x a Another way to say this is that the function f (x) f (a) F a (x) =
IMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION CALCULUS 3 INU0115/515 (MATHS 2) Dr Arian Jannetta MIMA CMath FRAS Implicit Differentiation 1/ 11 Arian Jannetta Explicit an implicit functions Explicit functions An explicit function
d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval
Exam 3 Review Lessons 17-18: Relative Extrema, Critical Numbers, an First Derivative Test (from exam 2 review neee for curve sketching) Critical Numbers: where the erivative of a function is zero or unefine.
Math 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x
. Fin erivatives of the following functions: (a) f() = tan ( 2 + ) ( ) 2 (b) f() = ln 2 + (c) f() = sin() Solution: Math 80, Eam 2, Fall 202 Problem Solution (a) The erivative is compute using the Chain
Differentiation Rules (2A) Young Won Lim 1/30/16
Differentiation Rules (2A) Copyright (c) 2011-2016 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
Math 106 Exam 2 Topics. du dx
The Chain Rule Math 106 Exam 2 Topics Composition (g f)(x 0 ) = g(f(x 0 )) ; (note: we on t know what g(x 0 ) is.) (g f) ought to have something to o with g (x) an f (x) in particular, (g f) (x 0 ) shoul
Introduction to ODE's (0P) Young Won Lim 12/27/14
Introuction to ODE's (0P) Copyright (c) 2011-2014 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
23 Implicit differentiation
23 Implicit ifferentiation 23.1 Statement The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x an y. If a value of x is given, then a corresponing value of y is etermine. For
A. Incorrect! The letter t does not appear in the expression of the given integral
AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question
Chapter 3 Differentiation Rules
Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number,
dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)
![dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)](/thumbs/75/71716329.jpg "dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)")
Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =
Implicit Differentiation and Inverse Trigonometric Functions
Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5
Computing Derivatives
Chapter 2 Computing Derivatives 2.1 Elementary erivative rules Motivating Questions In this section, we strive to unerstan the ieas generate by the following important questions: What are alternate notations
Short Review of Basic Mathematics
Short Review of Basic Mathematics Tomas Co January 1, 1 c 8 Tomas Co, all rights reserve Contents 1 Review of Functions 5 11 Mathematical Ientities 5 1 Drills 7 1 Answers to Drills 8 Review of ODE Solutions
Fall 2016: Calculus I Final
Answer the questions in the spaces provie on the question sheets. If you run out of room for an answer, continue on the back of the page. NO calculators or other electronic evices, books or notes are allowe
lim Prime notation can either be directly applied to a function as previously seen with f x 4.1 Basic Techniques for Finding Derivatives
MATH 040 Notes: Unit Page 4. Basic Techniques for Fining Derivatives In the previous unit we introuce the mathematical concept of the erivative: f f ( h) f ( ) lim h0 h (assuming the limit eists) In this
1 Applications of the Chain Rule
November 7, 08 MAT86 Week 6 Justin Ko Applications of the Chain Rule We go over several eamples of applications of the chain rule to compute erivatives of more complicate functions. Chain Rule: If z =
Section 2.7 Derivatives of powers of functions
Section 2.7 Derivatives of powers of functions (3/19/08) Overview: In this section we iscuss the Chain Rule formula for the erivatives of composite functions that are forme by taking powers of other functions.
QF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim
QF101: Quantitative Finance September 5, 2017 Week 3: Derivatives Facilitator: Christopher Ting AY 2017/2018 I recoil with ismay an horror at this lamentable plague of functions which o not have erivatives.
Chapter 12: Differentiation. SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M.
Chapter 12: Differentiation SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza Chapter 12: Differentiation Lecture 12.1: The Derivative Lecture
Lecture 4 : General Logarithms and Exponentials. a x = e x ln a, a > 0.
For a > 0 an x any real number, we efine Lecture 4 : General Logarithms an Exponentials. a x = e x ln a, a > 0. The function a x is calle the exponential function with base a. Note that ln(a x ) = x ln
CAUCHY INTEGRAL THEOREM
CAUCHY INTEGRAL THEOREM XI CHEN 1. Differential Forms, Integration an Stokes Theorem Let X be an open set in R n an C (X) be the set of complex value C functions on X. A ifferential 1-form is (1.1) ω =
Computing Derivatives Solutions
Stuent Stuy Session Solutions We have intentionally inclue more material than can be covere in most Stuent Stuy Sessions to account for groups that are able to answer the questions at a faster rate. Use
Strauss PDEs 2e: Section Exercise 6 Page 1 of 5
Strauss PDEs 2e: Section 4.3 - Exercise 6 Page 1 of 5 Exercise 6 If a 0 = a l = a in the Robin problem, show that: (a) There are no negative eigenvalues if a 0, there is one if 2/l < a < 0, an there are
x = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)
Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane