 3.1: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.2: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.3: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.4: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.5: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.6: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.7: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.8: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.9: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.10: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.11: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.12: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.13: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.14: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.15: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.16: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.17: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.18: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.19: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.20: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.21: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.22: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.23: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.24: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.25: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.26: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.27: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.28: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.29: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.30: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.31: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.32: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.33: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.34: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.35: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.36: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.37: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.38: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.39: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.40: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.41: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.42: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.43: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.44: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.45: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.46: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.47: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.48: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.49: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.50: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.51: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.52: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.53: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.54: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.55: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.56: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.57: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.58: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.59: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.60: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.61: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.62: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.63: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.64: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.65: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.66: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.67: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.68: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.69: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.70: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.71: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.72: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.73: Find derivatives for the functions in Exercises 173. Assumea, b, c,...
 3.74: For Exercises 7475, assume that y is a differentiable functionof x ...
 3.75: For Exercises 7475, assume that y is a differentiable functionof x ...
 3.76: Find the slope of the curve x2 + 3y2 = 7 at (2, 1).
 3.77: Assume y is a differentiable function of x and thaty+sin y+x2 = 9. ...
 3.78: Find the equations for the lines tangent to the graph ofxy + y2 = 4...
 3.79: If f(t)=2t 3 4t 2 + 3t 1, find f (t) and f(t).
 3.80: If f(x) = 13 8x + 2x2 and f (r)=4, find r.
 3.81: If f(x) = x3 6x2 15x + 20, find analytically allvalues of x for whi...
 3.82: (a) Find the eighth derivative of f(x) = x7 + 5x5 4x3 + 6x 7. Think...
 3.83: For 8388, use Figure 3.48.Let h(x) = t(x)s(x) and p(x) = t(x)/s(x)....
 3.84: For 8388, use Figure 3.48.Let r(x) = s(t(x)). Estimate r(0).
 3.85: For 8388, use Figure 3.48.Let h(x) = s(s(x)). Estimate:(a) h(1) (b)...
 3.86: For 8388, use Figure 3.48.Estimate all values of x for which the ta...
 3.87: For 8388, use Figure 3.48.Let h(x) = x2t(x) and p(x) = t(x2). Estim...
 3.88: For 8388, use Figure 3.48.Find an approximate equation for the tang...
 3.89: For 8992, let h(x) = f(g(x)) and k(x) =g(f(x)). Use Figure 3.49 to ...
 3.90: For 8992, let h(x) = f(g(x)) and k(x) =g(f(x)). Use Figure 3.49 to ...
 3.91: For 8992, let h(x) = f(g(x)) and k(x) =g(f(x)). Use Figure 3.49 to ...
 3.92: For 8992, let h(x) = f(g(x)) and k(x) =g(f(x)). Use Figure 3.49 to ...
 3.93: Using the information in the table about f and g, find:(a) h(4) if ...
 3.94: Given: r(2) = 4, s(2) = 1, s(4) = 2, r(2) = 1,s(2) = 3, s(4) = 3. C...
 3.95: If g(2) = 3 and g(2) = 4, find f(2) for the following:(a) f(x) = x2...
 3.96: For parts (a)(f) of 95, determine the equationof the line tangent t...
 3.97: Imagine you are zooming in on the graphs of the followingfunctions ...
 3.98: The graphs of sin x and cos x intersect once between 0and /2. What ...
 3.99: In 99100, show that the curves meet at least onceand determine whet...
 3.100: In 99100, show that the curves meet at least onceand determine whet...
 3.101: For some constant b and x > 0, let y = x ln xbx. Findthe equation o...
 3.102: In 102104, find the limit as x cosh(2x)sinh(3x)
 3.103: In 102104, find the limit as x e2xsinh(2x)
 3.104: In 102104, find the limit as x sinh(x2)cosh(x2)
 3.105: Consider the function f(x) = x.(a) Find and sketch f(x) and the tan...
 3.106: . Figure 3.50 shows the tangent line approximation to f(x)near x = ...
 3.107: Global temperatures have been rising, on average, formore than a ce...
 3.108: In 2009, the population of Hungary11 was approximatedbyP = 9.906(0....
 3.109: The gravitational attraction, F, between the earth and asatellite o...
 3.110: The distance, s, of a moving body from a fixed point isgiven as a f...
 3.111: At any time, t, a population, P(t), is growing at a rateproportiona...
 3.112: An object is oscillating at the end of a spring. Its position,in ce...
 3.113: The total number of people, N, who have contracted adisease by a ti...
 3.114: The world population was 6.7 billion at the beginning of2008. An ex...
 3.115: The acceleration due to gravity, g, is given byg = GMr2 ,where M is...
 3.116: Given that f and g are differentiable everywhere, g is theinverse o...
 3.117: An increasing function f(x) has the value f(10) = 5.Explain how you...
 3.118: A particle is moving on the xaxis. It has velocity v(x)when it is ...
 3.119: If f is decreasing and f(20) = 10, which of the followingmust be in...
 3.120: Find the nth derivative of the following functions:(a) ln x (b) xex...
 3.121: The derivative f gives the (absolute) rate of change of aquantity f...
 3.122: The relative rate of change of a function f is defined to bef/f. Fi...
 3.123: (a) Use a computer algebra system to differentiate(x + 1)x and (sin...
 3.124: For 124126,(a) Use a computer algebra system to find and simplify t...
 3.125: For 124126,(a) Use a computer algebra system to find and simplify t...
 3.126: For 124126,(a) Use a computer algebra system to find and simplify t...
Solutions for Chapter 3: SHORTCUTS TO DIFFERENTIATION
Full solutions for Calculus: Single and Multivariable  6th Edition
ISBN: 9780470888612
Solutions for Chapter 3: SHORTCUTS TO DIFFERENTIATION
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Calculus: Single and Multivariable was written by and is associated to the ISBN: 9780470888612. Chapter 3: SHORTCUTS TO DIFFERENTIATION includes 126 full stepbystep solutions. This textbook survival guide was created for the textbook: Calculus: Single and Multivariable , edition: 6. Since 126 problems in chapter 3: SHORTCUTS TO DIFFERENTIATION have been answered, more than 43430 students have viewed full stepbystep solutions from this chapter.

Additive inverse of a real number
The opposite of b , or b

Anchor
See Mathematical induction.

Blocking
A feature of some experimental designs that controls for potential differences between subject groups by applying treatments randomly within homogeneous blocks of subjects

Coefficient matrix
A matrix whose elements are the coefficients in a system of linear equations

Compounded monthly
See Compounded k times per year.

Decreasing on an interval
A function f is decreasing on an interval I if, for any two points in I, a positive change in x results in a negative change in ƒ(x)

Domain of validity of an identity
The set of values of the variable for which both sides of the identity are defined

Index
See Radical.

Index of summation
See Summation notation.

Initial value of a function
ƒ 0.

Inverse tangent function
The function y = tan1 x

Leading coefficient
See Polynomial function in x

Line graph
A graph of data in which consecutive data points are connected by line segments

Modulus
See Absolute value of a complex number.

Paraboloid of revolution
A surface generated by rotating a parabola about its line of symmetry.

Polar distance formula
The distance between the points with polar coordinates (r1, ?1 ) and (r2, ?2 ) = 2r 12 + r 22  2r1r2 cos 1?1  ?22

Product rule of logarithms
ogb 1RS2 = logb R + logb S, R > 0, S > 0,

Slopeintercept form (of a line)
y = mx + b

Sum of an infinite series
See Convergence of a series

Yscl
The scale of the tick marks on the yaxis in a viewing window.