 13.5.1: In Exercises 14, find using the appropriate Chain Rule.
 13.5.2: In Exercises 14, find using the appropriate Chain Rule.
 13.5.3: In Exercises 14, find using the appropriate Chain Rule.
 13.5.4: In Exercises 14, find using the appropriate Chain Rule.
 13.5.5: In Exercises 510, find (a) using the appropriate Chain Rule and (b)...
 13.5.6: In Exercises 510, find (a) using the appropriate Chain Rule and (b)...
 13.5.7: In Exercises 510, find (a) using the appropriate Chain Rule and (b)...
 13.5.8: In Exercises 510, find (a) using the appropriate Chain Rule and (b)...
 13.5.9: In Exercises 510, find (a) using the appropriate Chain Rule and (b)...
 13.5.10: In Exercises 510, find (a) using the appropriate Chain Rule and (b)...
 13.5.11: Projectile Motion In Exercises 11 and 12, the parametric equations ...
 13.5.12: Projectile Motion In Exercises 11 and 12, the parametric equations ...
 13.5.13: In Exercises 13 and 14, find using the appropriate Chain Rule. Eval...
 13.5.14: In Exercises 13 and 14, find using the appropriate Chain Rule. Eval...
 13.5.15: In Exercises 1518, find and using the appropriate Chain Rule, and e...
 13.5.16: In Exercises 1518, find and using the appropriate Chain Rule, and e...
 13.5.17: In Exercises 1518, find and using the appropriate Chain Rule, and e...
 13.5.18: In Exercises 1518, find and using the appropriate Chain Rule, and e...
 13.5.19: In Exercises 1922, find and (a) using the appropriate Chain Rule an...
 13.5.20: In Exercises 1922, find and (a) using the appropriate Chain Rule an...
 13.5.21: In Exercises 1922, find and (a) using the appropriate Chain Rule an...
 13.5.22: In Exercises 1922, find and (a) using the appropriate Chain Rule an...
 13.5.23: In Exercises 2326, find and by using the appropriate Chain Rule.
 13.5.24: In Exercises 2326, find and by using the appropriate Chain Rule.
 13.5.25: In Exercises 2326, find and by using the appropriate Chain Rule.
 13.5.26: In Exercises 2326, find and by using the appropriate Chain Rule.
 13.5.27: In Exercises 2730, differentiate implicitly to find
 13.5.28: In Exercises 2730, differentiate implicitly to find
 13.5.29: In Exercises 2730, differentiate implicitly to find
 13.5.30: In Exercises 2730, differentiate implicitly to find
 13.5.31: In Exercises 3138, differentiate implicitly to find the first parti...
 13.5.32: In Exercises 3138, differentiate implicitly to find the first parti...
 13.5.33: In Exercises 3138, differentiate implicitly to find the first parti...
 13.5.34: In Exercises 3138, differentiate implicitly to find the first parti...
 13.5.35: In Exercises 3138, differentiate implicitly to find the first parti...
 13.5.36: In Exercises 3138, differentiate implicitly to find the first parti...
 13.5.37: In Exercises 3138, differentiate implicitly to find the first parti...
 13.5.38: In Exercises 3138, differentiate implicitly to find the first parti...
 13.5.39: In Exercises 39 42, differentiate implicitly to find the first part...
 13.5.40: In Exercises 39 42, differentiate implicitly to find the first part...
 13.5.41: In Exercises 39 42, differentiate implicitly to find the first part...
 13.5.42: In Exercises 39 42, differentiate implicitly to find the first part...
 13.5.43: Homogeneous Functions In Exercises 43 46, the function is homogeneo...
 13.5.44: Homogeneous Functions In Exercises 43 46, the function is homogeneo...
 13.5.45: Homogeneous Functions In Exercises 43 46, the function is homogeneo...
 13.5.46: Homogeneous Functions In Exercises 43 46, the function is homogeneo...
 13.5.47: Let be a function where and are functions of a single variable Give...
 13.5.48: Let be a function where and are functions of two variables and Give...
 13.5.49: Describe the difference between the explicit form of a function of ...
 13.5.50: If give the rule for finding implicitly. If give the rule for findi...
 13.5.51: Volume and Surface Area The radius of a right circular cylinder is ...
 13.5.52: Volume and Surface Area Repeat Exercise 51 for a right circular cone.
 13.5.53: Area Let be the angle between equal sides of an isosceles triangle ...
 13.5.54: Volume and Surface Area The two radii of the frustum of a right cir...
 13.5.55: Moment of Inertia An annular cylinder has an inside radius of and a...
 13.5.56: Ideal Gas Law The Ideal Gas Law is where is a constant, is a consta...
 13.5.57: Maximum Angle A twofoottall painting hangs on a wall such that th...
 13.5.58: Show that if is homogeneous of degree then [Hint: Let Find and then...
 13.5.59: Show that for and
 13.5.60: Demonstrate the result of Exercise 59 for
 13.5.61: Consider the function where and Prove each of the following. (a) (b)
 13.5.62: Demonstrate the result of Exercise 61(b) for
 13.5.63: CauchyRiemann Equations Given the functions and verify that the Ca...
 13.5.64: Demonstrate the result of Exercise 63 for the functions and
Solutions for Chapter 13.5: Chain Rules for Functions of Several Variables
Full solutions for Calculus  8th Edition
ISBN: 9780618502981
Solutions for Chapter 13.5: Chain Rules for Functions of Several Variables
Get Full SolutionsCalculus was written by and is associated to the ISBN: 9780618502981. Since 64 problems in chapter 13.5: Chain Rules for Functions of Several Variables have been answered, more than 78032 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 13.5: Chain Rules for Functions of Several Variables includes 64 full stepbystep solutions. This textbook survival guide was created for the textbook: Calculus, edition: 8.

Absolute value of a vector
See Magnitude of a vector.

Arrow
The notation PQ denoting the directed line segment with initial point P and terminal point Q.

Compound interest
Interest that becomes part of the investment

Compounded monthly
See Compounded k times per year.

Directrix of a parabola, ellipse, or hyperbola
A line used to determine the conic

Focal width of a parabola
The length of the chord through the focus and perpendicular to the axis.

Limaçon
A graph of a polar equation r = a b sin u or r = a b cos u with a > 0 b > 0

Natural logarithm
A logarithm with base e.

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

Numerical derivative of ƒ at a
NDER f(a) = ƒ1a + 0.0012  ƒ1a  0.00120.002

Outcomes
The various possible results of an experiment.

Partial fractions
The process of expanding a fraction into a sum of fractions. The sum is called the partial fraction decomposition of the original fraction.

Random numbers
Numbers that can be used by researchers to simulate randomness in scientific studies (they are usually obtained from lengthy tables of decimal digits that have been generated by verifiably random natural phenomena).

Real zeros
Zeros of a function that are real numbers.

Rigid transformation
A transformation that leaves the basic shape of a graph unchanged.

RRAM
A Riemann sum approximation of the area under a curve ƒ(x) from x = a to x = b using x1 as the righthand end point of each subinterval.

Sine
The function y = sin x.

Stretch of factor c
A transformation of a graph obtained by multiplying all the xcoordinates (horizontal stretch) by the constant 1/c, or all of the ycoordinates (vertical stretch) of the points by a constant c, c, > 1.

symmetric about the xaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

zaxis
Usually the third dimension in Cartesian space.