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Calculus Volume 3

B | Table of Derivatives

Calculus Volume 3B | Table of Derivatives
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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

General Formulas

1. ddx(c)=0ddx(c)=0

2. ddx(f(x)+g(x))=f(x)+g(x)ddx(f(x)+g(x))=f(x)+g(x)

3. ddx(f(x)g(x))=f(x)g(x)+f(x)g(x)ddx(f(x)g(x))=f(x)g(x)+f(x)g(x)

4. ddx(xn)=nxn1,for real numbersnddx(xn)=nxn1,for real numbersn

5. ddx(cf(x))=cf(x)ddx(cf(x))=cf(x)

6. ddx(f(x)g(x))=f(x)g(x)ddx(f(x)g(x))=f(x)g(x)

7. ddx(f(x)g(x))=g(x)f(x)f(x)g(x)(g(x))2ddx(f(x)g(x))=g(x)f(x)f(x)g(x)(g(x))2

8. ddx[f(g(x))]=f(g(x))·g(x)ddx[f(g(x))]=f(g(x))·g(x)

Trigonometric Functions

9. ddx(sinx)=cosxddx(sinx)=cosx

10. ddx(tanx)=sec2xddx(tanx)=sec2x

11. ddx(secx)=secxtanxddx(secx)=secxtanx

12. ddx(cosx)=sinxddx(cosx)=sinx

13. ddx(cotx)=csc2xddx(cotx)=csc2x

14. ddx(cscx)=−cscxcotxddx(cscx)=−cscxcotx

Inverse Trigonometric Functions

15. ddx(sin−1x)=11x2ddx(sin−1x)=11x2

16. ddx(tan−1x)=11+x2ddx(tan−1x)=11+x2

17. ddx(sec−1x)=1|x|x21ddx(sec−1x)=1|x|x21

18. ddx(cos−1x)=11x2ddx(cos−1x)=11x2

19. ddx(cot−1x)=11+x2ddx(cot−1x)=11+x2

20. ddx(csc−1x)=1|x|x21ddx(csc−1x)=1|x|x21

Exponential and Logarithmic Functions

21. ddx(ex)=exddx(ex)=ex

22. ddx(ln|x|)=1xddx(ln|x|)=1x

23. ddx(bx)=bxlnbddx(bx)=bxlnb

24. ddx(logbx)=1xlnbddx(logbx)=1xlnb

Hyperbolic Functions

25. ddx(sinhx)=coshxddx(sinhx)=coshx

26. ddx(tanhx)=sech2xddx(tanhx)=sech2x

27. ddx(sechx)=−sechxtanhxddx(sechx)=−sechxtanhx

28. ddx(coshx)=sinhxddx(coshx)=sinhx

29. ddx(cothx)=csch2xddx(cothx)=csch2x

30. ddx(cschx)=−cschxcothxddx(cschx)=−cschxcothx

Inverse Hyperbolic Functions

31. ddx(sinh−1x)=1x2+1ddx(sinh−1x)=1x2+1

32. ddx(tanh−1x)=11x2(|x|<1)ddx(tanh−1x)=11x2(|x|<1)

33. ddx(sech−1x)=1x1x2(0<x<1)ddx(sech−1x)=1x1x2(0<x<1)

34. ddx(cosh−1x)=1x21(x>1)ddx(cosh−1x)=1x21(x>1)

35. ddx(coth−1x)=11x2(|x|>1)ddx(coth−1x)=11x2(|x|>1)

36. ddx(csch−1x)=1|x|1+x2(x0)ddx(csch−1x)=1|x|1+x2(x0)

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