A damped oscillator The displacement of a mass on a spring

Two equivalent forms of the Chain Rule for calculating the derivative of y = f 1g1x22 are presented in this section. State both forms.

Let h1x2 = f 1g1x22, where f and g are differentiable on their domains. If g112 = 3 and g_112 = 5, what else do you need to know to calculate h_112?

Fill in the blanks. The derivative of f 1g1x22 equals f _ evaluated at multiplied by g_ evaluated at .

Identify the inner and outer functions in the composition cos4 x.

Identify the inner and outer functions in the composition 1x2 + 102-5.

Express Q1x2 = cos4 1x2 + 12 as the composition of three functions; that is, identify f, g, and h so that Q1x2 = f 1g1h1x222.

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = 13x + 7210

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = 15x2 + 11x220

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = sin5 x

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = cos x5

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = e5x - 7

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = 17x - 1

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = 2x2 + 1

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = e1x

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = tan 5x2

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = sin x 4

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = sec ex

718. Version 1 of the Chain Rule Use Version 1 of the Chain Rule to calculate dy dx . y = e-x2

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = 13x2 + 7x210

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = 1x2 + 2x + 728

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = 110x + 1

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = 2x2 + 9

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = 517x3 + 12-3

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = cos 5t

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = sec 13x + 12

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = csc ex

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = tan ex

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = etan t

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = sin 14x3 + 3x + 12

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = csc 1t2 + t2

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = sin 121x2

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = cos4 u + sin4 u

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = 1sec x + tan x25

1934. Version 2 of the Chain Rule Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. y = sin 14 cos z2

3536. Similar-looking composite functions Two composite functions are given that look similar, but in fact are quite different. Identify the inner function u = g1x2 and the outer function y = f 1u2; then evaluate dy dx using the Chain Rule. a. y = cos3 x b. y = cos x3

3536. Similar-looking composite functions Two composite functions are given that look similar, but in fact are quite different. Identify the inner function u = g1x2 and the outer function y = f 1u2; then evaluate dy dx using the Chain Rule. a. y = 1ex23 b. y = e1x32

Chain Rule using a table Let h1x2 = f 1g1x22 and p1x2 = g1 f 1x22. Use the table to compute the following derivatives. a. h_132 b. h_122 c. p_142 d. p_122 e. h_152

Chain Rule using a table Let h1x2 = f 1g1x22 and k1x2 = g1g1x22. Use the table to compute the following derivatives. a. h_112 b. h_122 c. h_132 d. k_132 e. k_112 f. k_152 x 1 2 3 4 5 f_1x2 -6 -3 8 7 2 g1x2 4 1 5 2 3 g_1x2 9 7 3 -1 -5

Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.

Changing temperature The lapse rate is the rate at which the temperature in Earths atmosphere decreases with altitude. For example, a lapse rate of 6.5 Celsius/km means the temperature decreases at a rate of 6.5C per kilometer of altitude. The lapse rate varies with location and with other variables such as humidity. However, at a given time and location, the lapse rate is often nearly constant in the first 10 kilometers of the atmosphere. A radiosonde (weather balloon) is released from Earths surface, and its altitude (measured in km above sea level) at various times (measured in hours) is given in the table below. time (hr) 0 0.5 1 1.5 2 2.5 Altitude (km) 0.5 1.2 1.7 2.1 2.5 2.9 a. Assuming a lapse rate of 6.5 C>km, what is the approximate rate of change of the temperature with respect to time as the balloon rises 1.5 hours into the flight? Specify the units of your result, and use a forward difference quotient when estimating the required derivative. b. How does an increase in lapse rate change your answer in part (a)? c. Is it necessary to know the actual temperature to carry out the calculation in part (a)? Explain.

4144. Chain Rule for powers Use the Chain Rule to find the derivative of the following functions. y = 12x6 - 3x3 + 3225

4144. Chain Rule for powers Use the Chain Rule to find the derivative of the following functions. y = 1cos x + 2 sin x28

4144. Chain Rule for powers Use the Chain Rule to find the derivative of the following functions. y = 11 + 2 tan x215

4144. Chain Rule for powers Use the Chain Rule to find the derivative of the following functions. y = 11 - ex24

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = 21 + cot2 x

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = 213x - 422 + 3x

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = sin 1sin 1ex22

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = sin2 1e3x + 12

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = sin5 1cos 3x2

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = cos4 17x32

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = tan 1e13x2

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = 11 - e-0.05x2-1

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = 2x + 1x

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = 4x + 2x + 1x

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = f 1g1x222, where f and g are differentiable for all real numbers

4556. Repeated use of the Chain Rule Calculate the derivative of the following functions. y = 1 f 1g1xm222n, where f and g are differentiable for all real numbers, and m and n are integers

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = a x x + 1 b 5

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = a ex x + 1 b 8

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = ex2 + 1 sin x3

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = tan 1x ex2

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = u2 sec 5u

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = a 3x 4x + 2 b 5

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = 11x + 221x2 + 1224

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = e2x12x - 725

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = 2x4 + cos 2x

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = tet t + 1

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = 1p + p22 sin p2

5768. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. y = 1z + 423 tan z

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The function x sin x can be differentiated without using the Chain Rule. b. The function e2x + 1 should be differentiated using the Chain Rule. c. The derivative of a product is not the product of the derivatives, but the derivative of a composition is a product of derivatives. d. d dx P1Q1x22 = P_1x2Q_1x2.

7073. Second derivatives Find d2y dx2 for the following functions. y = x cos x2

7073. Second derivatives Find d2y dx2 for the following functions. y = sin x2

7073. Second derivatives Find d2y dx2 for the following functions. y = 2x2 + 2

7073. Second derivatives Find d2y dx2 for the following functions. y = e-2x2

Derivatives by different methods a. Calculate d dx 1x2 + x22 using the Chain Rule. Simplify your answer. b. Expand 1x2 + x22 first and then calculate the derivative. Verify that your answer agrees with part (a).

7576. Square root derivatives Find the derivative of the following functions. y = 1f 1x2, where f is differentiable and nonnegative at x

7576. Square root derivatives Find the derivative of the following functions. y = 1f 1x2g1x2, where f and g are differentiable and nonnegative at x

Tangent lines Determine an equation of the line tangent to the graph of y = 1x2 - 122 x3 - 6x - 1 at the point 13, 82. Graph the function and the tangent line.

Tangent lines Determine equations of the lines tangent to the graph of y = x25 - x2 at the points 11, 22 and 1-2, -22. Graph the function and the tangent lines.

Tangent lines Assume f and g are differentiable on their domains with h1x2 = f 1g1x22. Suppose the equation of the line tangent to the graph of g at the point 14, 72 is y = 3x - 5 and the equation of the line tangent to the graph of f at 17, 92 is y = -2x + 23. a. Calculate h142 and h_142. b. Determine an equation of the line tangent to the graph of h at the point on the graph where x = 4.

Tangent lines Assume f is a differentiable function whose graph passes through the point 11, 42. Suppose g1x2 = f 1x22 and the line tangent to the graph of f at 11, 42 is y = 3x + 1. Determine each of the following. a. g112 b. g_1x2 c. g_112 d. An equation of the line tangent to the graph of g when x = 1

Tangent lines Find the equation of the line tangent to y = e2x at x = 12 ln 3. Graph the function and the tangent line.

Composition containing sin x Suppose f is differentiable on 3-2, 24 with f _102 = 3 and f _112 = 5. Let g1x2 = f 1sin x2. Evaluate the following expressions. a. g_102 b. g_ap 2 b c. g_1p2

Composition containing sin x Suppose f is differentiable for all real numbers with f 102 = -3, f 112 = 3, f _102 = 3, and f _112 = 5. Let g1x2 = sin 1pf 1x22. Evaluate the following expressions. a. g_102 b. g_112

Use equation (2) to answer the following questions. a. Find dy dt , the velocity of the mass. Assume k and m are constant. b. How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring? c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness (k is increased by a factor of 4)? d. Assume that y has units of meters, t has units of seconds, m has units of kg, and k has units of kg>s2. Show that the units of the velocity in part (a) are consistent.

Use equation (2) to answer the following questions. a. Find the second derivative d2y dt2 . b. Verify that d2y dt2 = - k m y.

Use equation (2) to answer the following questions. a. The period T is the time required by the mass to complete one oscillation. Show that T = 2pA m k . b. Assume k is constant and calculate dT dm . c. Give a physical explanation of why dT dm is positive.

A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by y = 10e-t>2 cos pt 8 . a. Graph the displacement function. b. Compute and graph the velocity of the mass, v1t2 = y_1t2. c. Verify that the velocity is zero when the mass reaches the high and low points of its oscillation.

Oscillator equation A mechanical oscillator (such as a mass on a spring or a pendulum) subject to frictional forces satisfies the equation (called a differential equation) y_1t2 + 2y_1t2 + 5y1t2 = 0, where y is the displacement of the oscillator from its equilibrium position. Verify by substitution that the function y1t2 = e-t 1sin 2t - 2 cos 2t2 satisfies this equation.

Hours of daylight The number of hours of daylight at any point on Earth fluctuates throughout the year. In the northern hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40 north latitude, the length of a day is approximated by D1t2 = 12 - 3 cos a 2p1t + 102 365 b, where D is measured in hours and 0 t 365 is measured in days, with t = 0 corresponding to January 1. a. Approximately how much daylight is there on March 1 1t = 592? b. Find the rate at which the daylight function changes. c. Find the rate at which the daylight function changes on March 1. Convert your answer to units of min>day and explain what this result means. d. Graph the function y = D_1t2 using a graphing utility. e. At what times of year is the length of day changing most rapidly? Least rapidly?

A mixing tank A 500-liter (L) tank is filled with pure water. At time t = 0, a salt solution begins flowing into the tank at a rate of 5 L>min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L>min. The mass of salt in grams in the tank at any time t 0 is given by M1t2 = 25011000 - t211 - 10-3011000 - t2102 and the volume of solution in the tank (in liters) is given by V1t2 = 500 - 0.5t. a. Graph the mass function and verify that M102 = 0. b. Graph the volume function and verify that the tank is empty when t = 1000 min. c. The concentration of the salt solution in the tank (in g>L) is given by C1t2 = M1t2>V1t2. Graph the concentration function and comment on its properties. Specifically, what are C102 and lim tS1000- C1t2? d. Find the rate of change of the mass M_1t2, for 0 t 1000. e. Find the rate of change of the concentration C_1t2, for 0 t 1000. f. For what times is the concentration of the solution increasing? Decreasing?

Power and energy The total energy in megawatt-hr (MWh) used by a town is given by E1t2 = 400t + 2400 p sin pt 12 , where t 0 is measured in hours, with t = 0 corresponding to noon. a. Find the power, or rate of energy consumption, P1t2 = E_1t2 in units of megawatts (MW). b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day? c. At what time of day is the rate of energy consumption a minimum? What is the power at that time of day? d. Sketch a graph of the power function reflecting the times at which energy use is a minimum or maximum.

Deriving trigonometric identities a. Differentiate both sides of the identity cos 2t = cos2 t - sin2 t to prove that sin 2t = 2 sin t cos t. b. Verify that you obtain the same identity for sin 2t as in part (a) if you differentiate the identity cos 2t = 2 cos2 t - 1. c. Differentiate both sides of the identity sin 2t = 2 sin t cos t to prove that cos 2t = cos2 t - sin2 t.

Proof of cos2 x _ sin2 x _ 1 Let f 1x2 = cos2 x + sin2 x. a. Use the Chain Rule to show that f _1x2 = 0. b. Assume that if f _ = 0, then f is a constant function. Calculate f 102 and use it with part (a) to explain why cos2 x + sin2 x = 1.

Using the Chain Rule to prove that d dx 1ekx 2 _ kekx a. Identify the inner function g and the outer function f for the composition f 1g1x22 = ekx, where k is a real number. b. Use the Chain Rule to show that d dx 1ekx2 = kekx.

Deriving the Quotient Rule using the Product Rule and Chain Rule Suppose you forgot the Quotient Rule for calculating d dx a f 1x2 g1x2 b. Use the Chain Rule and Product Rule with the identity f 1x2 g1x2 = f 1x21g1x22-1 to derive the Quotient Rule.

The Chain Rule for second derivatives a. Derive a formula for the second derivative, d2 dx2 1 f 1g1x222. b. Use the formula in part (a) to calculate d2 dx2 1sin 13x4 + 5x2 + 222.

97100. Calculating limits The following limits are the derivatives of a composite function h at a point a. a. Find a possible function h and number a. b. Use the Chain Rule to find each limit. Verify your answer using a calculator. lim xS2 1x2 - 325 - 1 x - 2

97100. Calculating limits The following limits are the derivatives of a composite function h at a point a. a. Find a possible function h and number a. b. Use the Chain Rule to find each limit. Verify your answer using a calculator. lim xS0 14 + sin x - 2 x

97100. Calculating limits The following limits are the derivatives of a composite function h at a point a. a. Find a possible function h and number a. b. Use the Chain Rule to find each limit. Verify your answer using a calculator. lim hS0 sin 1p>2 + h22 - sin 1p2>42 h

97100. Calculating limits The following limits are the derivatives of a composite function h at a point a. a. Find a possible function h and number a. b. Use the Chain Rule to find each limit. Verify your answer using a calculator. lim hS0 1 3111 + h25 + 7210 - 1 318210 h

Limit of a difference quotient Assuming that f is differentiable for all x, simplify lim xS5 f 1x22 - f 1252 x - 5 .

Derivatives of even and odd functions Recall that f is even if f 1-x2 = f 1x2, for all x in the domain of f, and f is odd if f 1-x2 = -f 1x2, for all x in the domain of f. a. If f is a differentiable, even function on its domain, determine whether f _ is even, odd, or neither. b. If f is a differentiable, odd function on its domain, determine whether f _ is even, odd, or neither.

A general proof of the Chain Rule Let f and g be differentiable functions with h1x2 = f 1g1x22. For a given constant a, let u = g1a2 and v = g1x2, and define H1v2 = c f 1v2 - f 1u2 v - u - f _1u2 if v _ u 0 if v = u. a. Show that lim vSu H1v2 = 0. b. For any value of u show that f 1v2 - f 1u2 = 1H1v2 + f _1u221v - u2. c. Show that h_1a2 = lim xSa a 1H1g1x22 + f _1g1a222# g1x2 - g1a2 x - a b. d. Show that h_1a2 = f _1g1a22g_1a2.