 14.5.1: Use the Chain Rule to find dz/dt or dw/dt.
 14.5.2: Use the Chain Rule to find dz/dt or dw/dt.
 14.5.3: Use the Chain Rule to find dz/dt or dw/dt.
 14.5.4: Use the Chain Rule to find dz/dt or dw/dt.
 14.5.5: Use the Chain Rule to find dz/dt or dw/dt.
 14.5.6: Use the Chain Rule to find dz/dt or dw/dt.
 14.5.7: Use the Chain Rule to find zs and zt .
 14.5.8: Use the Chain Rule to find zs and zt .
 14.5.9: Use the Chain Rule to find zs and zt .
 14.5.10: Use the Chain Rule to find zs and zt .
 14.5.11: Use the Chain Rule to find zs and zt .
 14.5.12: Use the Chain Rule to find zs and zt .
 14.5.13: If , where is differentiable, and find when .
 14.5.14: Let , where are differentiable, and Find and .
 14.5.15: Suppose is a differentiable function of and , and . Use the table o...
 14.5.16: Suppose is a differentiable function of and , and Use the table of ...
 14.5.17: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.5.18: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.5.19: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.5.20: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.5.21: Use the Chain Rule to find the indicated partial derivatives.
 14.5.22: Use the Chain Rule to find the indicated partial derivatives.
 14.5.23: Use the Chain Rule to find the indicated partial derivatives.
 14.5.24: Use the Chain Rule to find the indicated partial derivatives.
 14.5.25: Use the Chain Rule to find the indicated partial derivatives.
 14.5.26: Use the Chain Rule to find the indicated partial derivatives.
 14.5.27: Use Equation 6 to find dy/dx.
 14.5.28: Use Equation 6 to find dy/dx.
 14.5.29: Use Equation 6 to find dy/dx.
 14.5.30: Use Equation 6 to find dy/dx.
 14.5.31: Use Equations 7 to find az/ax and az/ay .
 14.5.32: Use Equations 7 to find az/ax and az/ay .
 14.5.33: Use Equations 7 to find az/ax and az/ay .
 14.5.34: Use Equations 7 to find az/ax and az/ay .
 14.5.35: The temperature at a point is , measured in degrees Celsius. A bug ...
 14.5.36: Wheat production in a given year depends on the average temperature...
 14.5.37: The speed of sound traveling through ocean water with salinity 35 p...
 14.5.38: The radius of a right circular cone is increasing at a rate of in_s...
 14.5.39: The length _, width , and height of a box change with time. At a ce...
 14.5.40: The voltage in a simple electrical circuit is slowly decreasing as ...
 14.5.41: The pressure of 1 mole of an ideal gas is increasing at a rate of k...
 14.5.42: A manufacturer has modeled its yearly production function (the valu...
 14.5.43: One side of a triangle is increasing at a rate of and a second side...
 14.5.44: If a sound with frequency is produced by a source traveling along a...
 14.5.45: Assume that all the given functions are differentiable
 14.5.46: Assume that all the given functions are differentiable
 14.5.47: Assume that all the given functions are differentiable
 14.5.48: Assume that all the given functions are differentiable
 14.5.49: Assume that all the given functions have continuous secondorder pa...
 14.5.50: Assume that all the given functions have continuous secondorder pa...
 14.5.51: Assume that all the given functions have continuous secondorder pa...
 14.5.52: Assume that all the given functions have continuous secondorder pa...
 14.5.53: Assume that all the given functions have continuous secondorder pa...
 14.5.54: Assume that all the given functions have continuous secondorder pa...
 14.5.55: A function f is called homogeneous of degree n if it satisfies the ...
 14.5.56: If is homogeneous of degree , show that
 14.5.57: If is homogeneous of degree , show that
 14.5.58: Suppose that the equation implicitly defines each of the three vari...
 14.5.59: Equation 6 is a formula for the derivative of a function defined im...
Solutions for Chapter 14.5: THE CHAIN RULE
Full solutions for Multivariable Calculus,  7th Edition
ISBN: 9780538497879
Solutions for Chapter 14.5: THE CHAIN RULE
Get Full SolutionsSince 59 problems in chapter 14.5: THE CHAIN RULE have been answered, more than 23588 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Multivariable Calculus,, edition: 7. Multivariable Calculus, was written by and is associated to the ISBN: 9780538497879. Chapter 14.5: THE CHAIN RULE includes 59 full stepbystep solutions.

Binomial probability
In an experiment with two possible outcomes, the probability of one outcome occurring k times in n independent trials is P1E2 = n!k!1n  k2!pk11  p) nk where p is the probability of the outcome occurring once

Bounded above
A function is bounded above if there is a number B such that ƒ(x) ? B for all x in the domain of ƒ.

Complements or complementary angles
Two angles of positive measure whose sum is 90°

Compounded annually
See Compounded k times per year.

Compounded continuously
Interest compounded using the formula A = Pert

Cube root
nth root, where n = 3 (see Principal nth root),

Definite integral
The definite integral of the function ƒ over [a,b] is Lbaƒ(x) dx = limn: q ani=1 ƒ(xi) ¢x provided the limit of the Riemann sums exists

Dot product
The number found when the corresponding components of two vectors are multiplied and then summed

Focal length of a parabola
The directed distance from the vertex to the focus.

Implied domain
The domain of a function’s algebraic expression.

Increasing on an interval
A function ƒ is increasing on an interval I if, for any two points in I, a positive change in x results in a positive change in.

Mapping
A function viewed as a mapping of the elements of the domain onto the elements of the range

Natural logarithmic function
The inverse of the exponential function y = ex, denoted by y = ln x.

Negative association
A relationship between two variables in which higher values of one variable are generally associated with lower values of the other variable.

Quotient rule of logarithms
logb a R S b = logb R  logb S, R > 0, S > 0

Rational function
Function of the form ƒ(x)/g(x) where ƒ(x) and g(x) are polynomials and g(x) is not the zero polynomial.

Recursively defined sequence
A sequence defined by giving the first term (or the first few terms) along with a procedure for finding the subsequent terms.

Statistic
A number that measures a quantitative variable for a sample from a population.

Variation
See Power function.

yaxis
Usually the vertical coordinate line in a Cartesian coordinate system with positive direction up, pp. 12, 629.