Chain Rules for Functions of Several Variables

Slides:

Advertisements

Similar presentations

The Chain Rule Section 3.6c.

Advertisements

 Related Rates ◦ Idea:  Given two quantities that 1.Are somehow related 2.Changing (usually w.r.t. time)  The problem is to find how one of these quantities.

Implicit Differentiation

Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

© 2010 Pearson Education, Inc. All rights reserved.

2.5 Implicit Differentiation. Implicit and Explicit Functions Explicit FunctionImplicit Function But what if you have a function like this…. To differentiate:

Functions of Several Variables Copyright © Cengage Learning. All rights reserved.

MAT 1236 Calculus III Section 14.5 The Chain Rule

Section 2.5 – Implicit Differentiation

Chapter 14 – Partial Derivatives

Implicit Differentiation 3.6. Implicit Differentiation So far, all the equations and functions we looked at were all stated explicitly in terms of one.

Section 2.5 – Implicit Differentiation. Explicit Equations The functions that we have differentiated and handled so far can be described by expressing.

Implicit Differentiation

Section 2.5 Implicit Differentiation

Ms. Battaglia AB/BC Calculus. Up to this point, most functions have been expressed in explicit form. Ex: y=3x 2 – 5 The variable y is explicitly written.

Warm-Up: Find f’(x) if f(x)=(3x 2 -6x+2) 3. SECTION 6.4: IMPLICIT DIFFERENTIATION Objective: Students will be able to…  Take the derivative of implicitly.

Warm Up. Turn in chain rule HW Implicit Differentiation – 4.2.

5.4 Fundamental Theorem of Calculus. It is difficult to overestimate the power of the equation: It says that every continuous function f is the derivative.

Section 3.4 The Chain Rule. Consider the function –We can “decompose” this function into two functions we know how to take the derivative of –For example.

2.4: THE CHAIN RULE. Review: Think About it!!  What is a derivative???

Derivative of Exponential Function and Logarithms I Logarithms, still.

The Chain Rule. The Chain Rule Case I z x y t t start with z z is a function of x and y x and y are functions of t Put the appropriate derivatives along.

Aim: How do we take second derivatives implicitly? Do Now: Find the slope or equation of the tangent line: 1)3x² - 4y² + y = 9 at (2,1) 2)2x – 5y² = -x.

Chapter 3 Derivatives.

Derivatives of exponentials and Logarithms

AP Calculus BC September 12, 2016.

Section 14.2 Computing Partial Derivatives Algebraically

Chapter 14 Partial Derivatives.

6.1 – 6.3 Differential Equations

Implicit Differentiation

Copyright © Cengage Learning. All rights reserved.

Implicit Differentiation

4.2 – Implicit Differentiation

Section 3.7 Implicit Functions

3.6 Warm-Up Find y´´ Find the Derivative:.

Implicit Differentiation

Chain Rules for Functions of Several Variables

MTH1170 Implicit Differentiation

Implicit Differentiation Implicit differentiation

Calculus I (MAT 145) Dr. Day Wednesday Oct 11, 2017

DIFFERENTIATION & INTEGRATION

Copyright © Cengage Learning. All rights reserved.

Section 15.5 The Chain Rule.

Sec 3.5: IMPLICIT DIFFERENTIATION

4.2 – Implicit Differentiation

Implicit Differentiation

Differentiation and integration part 1

Chapter 3 Derivatives.

2.5 Implicit Differentiation

Chain Rule AP Calculus.

Implicit Differentiation

AP Calculus Honors Ms. Olifer

Implicit Differentiation

Chapter 4 More Derivatives Section 4.1 Chain Rule.

13 Functions of Several Variables

4.5 Integration by Substitution The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for.

Rates that Change Based on another Rate Changing

Implicit Differentiation

Implicit Differentiation

Functions of Several Variables

Differentiate the function: {image} .

Derivatives of Inverse Trig Functions

14.4 Chain Rules for functions of several variables

3.5 The Chain Rule Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002.

The Chain Rule Section 3.6b.

Chapter 4 More Derivatives Section 4.1 Chain Rule.

Implicit Differentiation

Section 5.5: The Substitution Rule

The Chain Rule.

Presentation transcript:

Chain Rules for Functions of Several Variables

1. Find dw/dt using the appropriate Chain Rule (Similar to p.931 #1-4)

2. Find dw/dt (a) by using the appropriate Chain Rule, and (b) by converting w to a function of t before differentiating (Similar to p.931 #5-10)

3. Find dw/dt (a) by using the appropriate Chain Rule, and (b) by converting w to a function of t before differentiating (Similar to p.931 #5-10)

4. Find d2w/dt2 using the appropriate Chain Rule 4. Find d2w/dt2 using the appropriate Chain Rule. Evaluate d2w/dt2 at the given value of t (Similar to p.931 #13-14)

5. Find ∂w/∂s and ∂w/∂t using the appropriate Chain Rule and evaluate each partial derivative at the given values of s and t (Similar to p.931 #15-18)

6. Find ∂w/∂r and ∂w/∂θ (a) by using the appropriate Chain Rule and (b) by converting w to a function of r and θ before differentiating (Similar to p.931 #19-22)

7. Find ∂w/∂s and ∂w/∂t by using the appropriate Chain Rule (Similar to p.931 #23-26)

8. Differentiate implicitly to find dy/dx (Similar to p.931 #27-30)

9. Differentiate implicitly to find dy/dx (Similar to p.931 #27-30)

10. Differentiate implicitly to find the first partial derivatives of z (Similar to p.931 #31-38)

11. Differentiate implicitly to find the first partial derivatives of z (Similar to p.931 #31-38)

12. Differentiate implicitly to find the first partial derivatives of w (Similar to p.931 #39-42)