 3.7.1: Identify the outside and inside functions for each of these composi...
 3.7.2: Which of the following can be differentiated easily without using t...
 3.7.3: Which is the derivative of f (5x)? (a) 5f (x) (b) 5f (5x) (c) f (5x...
 3.7.4: Suppose that f (4) = g(4) = g (4) = 1. Do we have enough informatio...
 3.7.5: In Exercises 5 and 6, write the function as a composite f (g(x)) an...
 3.7.6: In Exercises 5 and 6, write the function as a composite f (g(x)) an...
 3.7.7: Calculate d dx cos u for the following choices of u(x): (a) u(x) = ...
 3.7.8: Calculate d dx f (x2 + 1) for the following choices of f (u): (a) f...
 3.7.9: Compute df dx if df du = 2 and du dx = 6.
 3.7.10: Compute df dx x=2 if f (u) = u2, u(2) = 5, and u (2) = 5.
 3.7.11: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.12: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.13: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.14: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.15: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.16: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.17: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.18: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.19: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.20: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.21: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.22: In Exercises 1122, use the General Power Rule, Exponential Rule, or...
 3.7.23: In Exercises 2326, compute the derivative of f g. f (u) = sin u, g(...
 3.7.24: In Exercises 2326, compute the derivative of f g. f (u) = 2u + 1, g...
 3.7.25: In Exercises 2326, compute the derivative of f g. f (u) = eu, g(x) ...
 3.7.26: In Exercises 2326, compute the derivative of f g. f (u) = u u 1 , g...
 3.7.27: In Exercises 27 and 28, find the derivatives of f (g(x)) and g(f (x...
 3.7.28: In Exercises 27 and 28, find the derivatives of f (g(x)) and g(f (x...
 3.7.29: In Exercises 2942, use the Chain Rule to find the derivative. y = s...
 3.7.30: In Exercises 2942, use the Chain Rule to find the derivative. y = s...
 3.7.31: In Exercises 2942, use the Chain Rule to find the derivative. y = t...
 3.7.32: In Exercises 2942, use the Chain Rule to find the derivative. y = t...
 3.7.33: In Exercises 2942, use the Chain Rule to find the derivative. y = (...
 3.7.34: In Exercises 2942, use the Chain Rule to find the derivative. y = (...
 3.7.35: In Exercises 2942, use the Chain Rule to find the derivative. y = x...
 3.7.36: In Exercises 2942, use the Chain Rule to find the derivative. y = c...
 3.7.37: In Exercises 2942, use the Chain Rule to find the derivative. y = s...
 3.7.38: In Exercises 2942, use the Chain Rule to find the derivative. y = t...
 3.7.39: In Exercises 2942, use the Chain Rule to find the derivative. y = t...
 3.7.40: In Exercises 2942, use the Chain Rule to find the derivative. y = e2x2
 3.7.41: In Exercises 2942, use the Chain Rule to find the derivative. y = e...
 3.7.42: In Exercises 2942, use the Chain Rule to find the derivative. y = c...
 3.7.43: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.44: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.45: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.46: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.47: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.48: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.49: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.50: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.51: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.52: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.53: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.54: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.55: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.56: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.57: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.58: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.59: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.60: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.61: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.62: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.63: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.64: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.65: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.66: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.67: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.68: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.69: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.70: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.71: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.72: In Exercises 4372, find the derivative using the appropriate rule o...
 3.7.73: In Exercises 7376, compute the higher derivative d2 dx2 sin(x2)
 3.7.74: In Exercises 7376, compute the higher derivative d2 dx2 (x2 + 9) 5
 3.7.75: In Exercises 7376, compute the higher derivative d3 dx3 (9 x)8
 3.7.76: In Exercises 7376, compute the higher derivative d3 dx3 sin(2x)
 3.7.77: The average molecular velocity v of a gas in a certain container is...
 3.7.78: The power P in a circuit is P = Ri2, where R is the resistance and ...
 3.7.79: An expanding sphere has radius r = 0.4t cm at time t (in seconds). ...
 3.7.80: A 2005 study by the Fisheries Research Services in Aberdeen, Scotla...
 3.7.81: A 1999 study by Starkey and Scarnecchia developed the following mod...
 3.7.82: The functions in Exercises 80 and 81 are examples of the von Bertal...
 3.7.83: With notation as in Example 8, calculate (a) d d sin =60 (b) d d + ...
 3.7.84: With notation as in Example 8, calculate (a) d d sin =60 (b) d d + ...
 3.7.85: Compute the derivative of h(sin x) at x = 6 , assuming that h (0.5)...
 3.7.86: Let F (x) = f (g(x)), where the graphs of f and g are shown in Figu...
 3.7.87: In Exercises 8790, use the table of values to calculate the derivat...
 3.7.88: In Exercises 8790, use the table of values to calculate the derivat...
 3.7.89: In Exercises 8790, use the table of values to calculate the derivat...
 3.7.90: In Exercises 8790, use the table of values to calculate the derivat...
 3.7.91: The price (in dollars) of a computer component is P = 2C 18C1, wher...
 3.7.92: Plot the astroid y = (4 x2/3)3/2 for 0 x 8. Show that the part of e...
 3.7.93: According to the U.S. standard atmospheric model, developed by the ...
 3.7.94: Climate scientists use the StefanBoltzmann Law R = T 4 to estimate ...
 3.7.95: In the setting of Exercise 94, calculate the yearly rate of change ...
 3.7.96: Use a computer algebra system to compute f (k)(x) for k = 1, 2, 3 f...
 3.7.97: Use the Chain Rule to express the second derivative of f g in terms...
 3.7.98: Compute the second derivative of sin(g(x)) at x = 2, assuming that ...
 3.7.99: Show that if f , g, and h are differentiable, then [f (g(h(x)))] = ...
 3.7.100: Show that differentiation reverses parity: If f is even, then f is ...
 3.7.101: (a) Sketch a graph of any even function f and explain graphically w...
 3.7.102: Power Rule for Fractional Exponents Let f (u) = uq and g(x) = xp/q ...
 3.7.103: Prove that for all whole numbers n 1, dn dxn sin x = sin x + n 2 Hi...
 3.7.104: A Discontinuous Derivative Use the limit definition to show that g ...
 3.7.105: Chain Rule This exercise proves the Chain Rule without the special ...
Solutions for Chapter 3.7: The Chain Rule
Full solutions for Calculus: Early Transcendentals  3rd Edition
ISBN: 9781464114885
Solutions for Chapter 3.7: The Chain Rule
Get Full SolutionsCalculus: Early Transcendentals was written by and is associated to the ISBN: 9781464114885. Chapter 3.7: The Chain Rule includes 105 full stepbystep solutions. Since 105 problems in chapter 3.7: The Chain Rule have been answered, more than 40237 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals , edition: 3. This expansive textbook survival guide covers the following chapters and their solutions.

Absolute value of a real number
Denoted by a, represents the number a or the positive number a if a < 0.

Angle between vectors
The angle formed by two nonzero vectors sharing a common initial point

Bounded interval
An interval that has finite length (does not extend to ? or ?)

Combinatorics
A branch of mathematics related to determining the number of elements of a set or the number of ways objects can be arranged or combined

Commutative properties
a + b = b + a ab = ba

Cosecant
The function y = csc x

Cycloid
The graph of the parametric equations

Dependent variable
Variable representing the range value of a function (usually y)

Equilibrium price
See Equilibrium point.

Feasible points
Points that satisfy the constraints in a linear programming problem.

Horizontal component
See Component form of a vector.

Mean (of a set of data)
The sum of all the data divided by the total number of items

Opposite
See Additive inverse of a real number and Additive inverse of a complex number.

Parameter interval
See Parametric equations.

Position vector of the point (a, b)
The vector <a,b>.

Real number line
A horizontal line that represents the set of real numbers.

Removable discontinuity at x = a
lim x:a ƒ(x) = limx:a+ ƒ(x) but either the common limit is not equal ƒ(a) to ƒ(a) or is not defined

Union of two sets A and B
The set of all elements that belong to A or B or both.

Vertical component
See Component form of a vector.

Zoom out
A procedure of a graphing utility used to view more of the coordinate plane (used, for example, to find theend behavior of a function).