Higher-order derivatives Find and simplify y''.

y = 2^x x

72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

9–61. Evaluate and simplify y'.

y = tan^−1 √t²−1

9–61. Evaluate and simplify y'.

y = 10^sin x+sin¹⁰x

9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x

9–61. Evaluate and simplify y'.

y = (sin x / cos x+1)^1/3

9–61. Evaluate and simplify y'.

y = 5t² sin t

Evaluate and simplify y'.

xy⁴+x⁴y=1

66–71. Higher-order derivatives Find and simplify y''.

x + sin y = y

9–61. Evaluate and simplify y'.

y = e^sin (cosx)

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

b. d/dx ((f(x) / g(x)) |x=

Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

y =√x³+x−1 at y=3

A jet flying at 450 mi/hr and traveling in a straight line at a constant elevation of 500 ft passes directly over a spectator at an air show. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later? 

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

a. Find equations of all lines tangent to the curve at the given value of x.

x+y³−y=1; x=1

A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e^-t cos t), for t ≥ 0.

Determine her velocity at t = 1 and t = 3. 

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.

a. Find db/da for a torus with a volume of 64π².

45–50. Tangent lines Carry out the following steps. <IMAGE>

a. Verify that the given point lies on the curve.

x⁴-x²y+y⁴=1; (−1, 1)

Comparing velocities Two stones are thrown vertically upward, each with an initial velocity of 48 ft/s at time t=0. One stone is thrown from the edge of a bridge that is 32 feet above the ground, and the other stone is thrown from ground level. The height above the ground of the stone thrown from the bridge after t seconds is f(t) = − 16t²+48t+32. and the height of the stone thrown from the ground after t seconds is g(t) = −16t²+48t.

a. Show that the stones reach their high points at the same time.

Vertical​ ​tangent lines

a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

g(s) = 4s³ - 8s² +4s / 4s

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>

a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)

x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)

{Use of Tech} Flow from a tank A cylindrical tank ​​is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

a. Graph the volume function. What is the volume of water in the tank before the valve is opened? 

Find an equation of the line tangent to the following curves at the given value of x.

y = 4 sin x cos x; x = π/3

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

a. Compute A'(t). What units are associated with this derivative and what does the derivative measure?

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = x² - 4; P(2, 0)

{Use of Tech} Tree growth ​Let​ ​b ​​represent ​the base diameter of a conifer tree and ​let ​​h ​represent the height of the tree, where ​b ​is measured in centimeters ​and ​​h ​is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².

a. Graph the height function.

Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. <IMAGE>

a. At which points is the slope of the curve negative?

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

f(w) = w³ -w / w

Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>

a. Determine the average velocity of the car during the first 45 minutes of the trip.

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

x = e^y; (2, ln 2)