- 3.4.1: ?Decomposition of a Composite Function In Exercises 1–6, complete t...
- 3.4.2: ?Decomposition of a Composite Function In Exercises 1–6, complete t...
- 3.4.3: ?Decomposition of a Composite Function In Exercises 1–6, complete t...
- 3.4.4: ?Decomposition of a Composite Function In Exercises 1–6, complete t...
- 3.4.5: ?Decomposition of a Composite Function In Exercises 1–6, complete t...
- 3.4.6: ?Decomposition of a Composite Function In Exercises 1–6, complete t...
- 3.4.7: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.8: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.9: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.10: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.11: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.12: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.13: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.14: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.15: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.16: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.17: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.18: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.19: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.20: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.21: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.22: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.23: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.24: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.25: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.26: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.27: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.28: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.29: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.30: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.31: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.32: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.33: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.34: ?Finding a Derivative In Exercises 7–34, find the derivative of the...
- 3.4.35: ?Finding a Derivative Using Technology In Exercises 35–40, use a co...
- 3.4.36: ?Finding a Derivative Using Technology In Exercises 35–40, use a co...
- 3.4.37: ?Finding a Derivative Using Technology In Exercises 35–40, use a co...
- 3.4.38: ?Finding a Derivative Using Technology In Exercises 35–40, use a co...
- 3.4.39: ?Finding a Derivative Using Technology In Exercises 35–40, use a co...
- 3.4.40: ?Finding a Derivative Using Technology In Exercises 35–40, use a co...
- 3.4.41: ?Slope of a Tangent Line In Exercises 41 and 42, find the slope of ...
- 3.4.42: ?Slope of a Tangent Line In Exercises 41 and 42, find the slope of ...
- 3.4.43: ?Slope of a Tangent Line In Exercises 43–46, find the slope of the ...
- 3.4.44: ?Slope of a Tangent Line In Exercises 43–46, find the slope of the ...
- 3.4.45: ?Slope of a Tangent Line In Exercises 43–46, find the slope of the ...
- 3.4.46: ?Slope of a Tangent Line In Exercises 43–46, find the slope of the ...
- 3.4.47: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.48: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.49: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.50: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.51: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.52: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.53: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.54: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.55: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.56: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.57: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.58: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.59: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.60: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.61: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.62: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.63: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.64: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.65: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.66: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.67: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.68: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.69: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.70: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.71: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.72: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.73: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.74: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.75: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.76: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.77: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.78: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.79: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.80: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.81: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.82: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.83: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.84: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.85: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.86: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.87: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.88: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.89: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.90: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.91: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.92: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.93: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.94: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.95: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.96: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.97: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.98: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.99: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.100: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.101: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.102: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.103: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.104: ?Finding a Derivative In Exercises 47– 104, find the derivative of ...
- 3.4.105: ?Evaluating a Derivative In Exercises 105–112, find and evaluate th...
- 3.4.106: ?Evaluating a Derivative In Exercises 105–112, find and evaluate th...
- 3.4.107: ?Evaluating a Derivative In Exercises 105–112, find and evaluate th...
- 3.4.108: ?Evaluating a Derivative In Exercises 105–112, find and evaluate th...
- 3.4.109: ?Evaluating a Derivative In Exercises 105–112, find and evaluate th...
- 3.4.110: ?Evaluating a Derivative In Exercises 105–112, find and evaluate th...
- 3.4.111: ?Evaluating a Derivative In Exercises 105–112, find and evaluate th...
- 3.4.112: ?Evaluating a Derivative In Exercises 105–112, find and evaluate th...
- 3.4.113: ?Finding an Equation of a Tangent Line In Exercises 113–120, (a) fi...
- 3.4.114: ?Finding an Equation of a Tangent Line In Exercises 113–120, (a) fi...
- 3.4.115: ?Finding an Equation of a Tangent Line In Exercises 113–120, (a) fi...
- 3.4.116: ?Finding an Equation of a Tangent Line In Exercises 113–120, (a) fi...
- 3.4.117: ?Finding an Equation of a Tangent Line In Exercises 113–120, (a) fi...
- 3.4.118: ?Finding an Equation of a Tangent Line In Exercises 113–120, (a) fi...
- 3.4.119: ?Finding an Equation of a Tangent Line In Exercises 113–120, (a) fi...
- 3.4.120: ?Finding an Equation of a Tangent Line In Exercises 113–120, (a) fi...
- 3.4.121: ?Famous Curves In Exercises 121 and 122, find an equation of the ta...
- 3.4.122: ?Famous Curves In Exercises 121 and 122, find an equation of the ta...
- 3.4.123: ?Horizontal Tangent Line Determine the point(s) in the interval \((...
- 3.4.124: ?Horizontal Tangent Line Determine the point(s) at which the graph ...
- 3.4.125: ?Finding a Second Derivative In Exercises 125–132, find the second ...
- 3.4.126: ?Finding a Second Derivative In Exercises 125–132, find the second ...
- 3.4.127: ?Finding a Second Derivative In Exercises 125–132, find the second ...
- 3.4.128: ?Finding a Second Derivative In Exercises 125–132, find the second ...
- 3.4.129: ?Finding a Second Derivative In Exercises 125–132, find the second ...
- 3.4.130: ?Finding a Second Derivative In Exercises 125–132, find the second ...
- 3.4.131: ?Finding a Second Derivative In Exercises 125–132, find the second ...
- 3.4.132: ?Finding a Second Derivative In Exercises 125–132, find the second ...
- 3.4.133: ?Evaluating a Second Derivative In Exercises 133–136, evaluate the ...
- 3.4.134: ?Evaluating a Second Derivative In Exercises 133–136, evaluate the ...
- 3.4.135: ?Evaluating a Second Derivative In Exercises 133–136, evaluate the ...
- 3.4.136: ?Evaluating a Second Derivative In Exercises 133–136, evaluate the ...
- 3.4.137: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.138: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.139: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.140: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.141: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.142: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.143: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.144: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.145: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.146: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.147: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.148: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.149: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.150: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.151: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.152: ?Finding a Derivative In Exercises 137–152, find the derivative of ...
- 3.4.153: ?Identifying Graphs In Exercises 153–156, the graphs of a function ...
- 3.4.154: ?Identifying Graphs In Exercises 153–156, the graphs of a function ...
- 3.4.155: ?Identifying Graphs In Exercises 153–156, the graphs of a function ...
- 3.4.156: ?Identifying Graphs In Exercises 153–156, the graphs of a function ...
- 3.4.157: ?Describing a Relationship In Exercises 157 and 158, the relationsh...
- 3.4.158: ?Describing a Relationship In Exercises 157 and 158, the relationsh...
- 3.4.159: ?Using Relationships In Exercises 159–162, given that \(g(5)=-3, \q...
- 3.4.160: ?Using Relationships In Exercises 159–162, given that \(g(5)=-3, \q...
- 3.4.161: ?Using Relationships In Exercises 159–162, given that \(g(5)=-3, \q...
- 3.4.162: ?Using Relationships In Exercises 159–162, given that \(g(5)=-3, \q...
- 3.4.163: ?Finding Derivatives In Exercises 163 and 164, the graphs of f and ...
- 3.4.164: ?Finding Derivatives In Exercises 163 and 164, the graphs of f and ...
- 3.4.165: ?Doppler Effect The frequency F of a fire truck siren heard by a st...
- 3.4.166: ?Harmonic Motion The displacement from equilibrium of an object in ...
- 3.4.167: ?Pendulum A 15-centimeter pendulum moves according to the equation ...
- 3.4.168: ?Wave Motion A buoy oscillates in simple harmonic motion \(y=A \cos...
- 3.4.169: ?Modeling Data The normal daily maximum temperatures T (in degrees ...
- 3.4.170: ?How Do You See It? The cost C (in dollars) of producing x units of...
- 3.4.171: ?Modeling Data The table shows the temperatures T (oF) at which wat...
- 3.4.172: ?Think About It The table shows some values of the derivative of an...
- 3.4.173: ?Circulatory System The speed S of blood that is r centimeters from...
- 3.4.174: ?Inflation If the annual rate of inflation averages 5% over the nex...
- 3.4.175: ?Biology The number N of bacteria in a culture after days is modele...
- 3.4.176: Depreciation The value V of a machine years after it is purchased i...
- 3.4.177: ?Finding a Pattern Consider the function \(f(x)=\sin \beta x\), whe...
- 3.4.178: ?Conjecture Let be a differentiable function of period p.(a) Is the...
- 3.4.179: ?Think About It Let \(r(x)=f(g(x))\) and \(s(x)=g(f(x)), where f an...
- 3.4.180: ?Using Trigonometric Functions(a) Find the derivative of the functi...
- 3.4.181: ?Even and Odd Functions(a) Show that the derivative of an odd funct...
- 3.4.182: ?Proof Let be a differentiable function of x. Use the fact that \(|...
- 3.4.183: ?Using Absolute Value In Exercises 183–186, use the result of Exerc...
- 3.4.184: ?Using Absolute Value In Exercises 183–186, use the result of Exerc...
- 3.4.185: ?Using Absolute Value In Exercises 183–186, use the result of Exerc...
- 3.4.186: ?Using Absolute Value In Exercises 183–186, use the result of Exerc...
- 3.4.187: ?Linear and Quadratic Approximations The linear and quadratic appro...
- 3.4.188: ?Linear and Quadratic Approximations The linear and quadratic appro...
- 3.4.189: ?Linear and Quadratic Approximations The linear and quadratic appro...
- 3.4.190: ?Linear and Quadratic Approximations The linear and quadratic appro...
- 3.4.191: ?True or False? In Exercises 191–194, determine whether the stateme...
- 3.4.192: ?True or False? In Exercises 191–194, determine whether the stateme...
- 3.4.193: True or False? In Exercises 191–194, determine whether the statemen...
- 3.4.194: ?True or False? In Exercises 191–194, determine whether the stateme...
- 3.4.195: ?Let \(f(x)=a_{1} \sin x+a_{2} \sin 2 x+\cdots+a_{n} \sin n x\), wh...
- 3.4.196: ?Let k be a fixed positive integer. The th derivative of \(\frac{1}...
Solutions for Chapter 3.4: The Chain Rule
Full solutions for Calculus: Early Transcendental Functions | 6th Edition
ISBN: 9781285774770
This textbook survival guide was created for the textbook: Calculus: Early Transcendental Functions, edition: 6. Calculus: Early Transcendental Functions was written by and is associated to the ISBN: 9781285774770. Chapter 3.4: The Chain Rule includes 196 full step-by-step solutions. Since 196 problems in chapter 3.4: The Chain Rule have been answered, more than 184376 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.
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Augmented matrix
A matrix that represents a system of equations.
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Circular functions
Trigonometric functions when applied to real numbers are circular functions
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Commutative properties
a + b = b + a ab = ba
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Difference identity
An identity involving a trigonometric function of u - v
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Distance (on a number line)
The distance between real numbers a and b, or |a - b|
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Hypotenuse
Side opposite the right angle in a right triangle.
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Imaginary part of a complex number
See Complex number.
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Interval notation
Notation used to specify intervals, pp. 4, 5.
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Leading term
See Polynomial function in x.
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Linear system
A system of linear equations
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Minor axis
The perpendicular bisector of the major axis of an ellipse with endpoints on the ellipse.
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n factorial
For any positive integer n, n factorial is n! = n.(n - 1) . (n - 2) .... .3.2.1; zero factorial is 0! = 1
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Negative numbers
Real numbers shown to the left of the origin on a number line.
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Octants
The eight regions of space determined by the coordinate planes.
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Power regression
A procedure for fitting a curve y = a . x b to a set of data.
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Probability distribution
The collection of probabilities of outcomes in a sample space assigned by a probability function.
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Range (in statistics)
The difference between the greatest and least values in a data set.
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Reflection through the origin
x, y and (-x,-y) are reflections of each other through the origin.
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Reflexive property of equality
a = a
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Upper bound test for real zeros
A test for finding an upper bound for the real zeros of a polynomial.