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Problem 3.R.67
Higher-order derivatives Find and simplify y''.
y = 2^x x
Problem 3.R.29
9–61. Evaluate and simplify y'.
y = tan^−1 √t²−1
Problem 3.R.63
Evaluate d/dx(x sec^−1 x) |x = 2 /√3.
Problem 3.R.48
9–61. Evaluate and simplify y'.
y = 10^sin x+sin¹⁰x
Problem 3.R.17
9–61. Evaluate and simplify y'.
y = 5t² sin t
Problem 3.R.73
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)
Problem 3.R.54
9–61. Evaluate and simplify y'.
y = 2x² cos^−1 x+ sin^−1 x
Problem 3.R.56
Evaluate and simplify y'.
xy⁴+x⁴y=1
Problem 3.R.40
9–61. Evaluate and simplify y'.
y = e^sin (cosx)
Problem 3.R.32
9–61. Evaluate and simplify y'.
y = csc⁵ 3x
Problem 3.R.65
Find f′(1) when f(x) = x^(1/x).
Problem 3.R.85b
Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
b. d/dx ((f(x) / g(x)) |x=
Problem 3.R.33
Evaluate and simplify y'.
y = ln w / w⁵
Problem 3.6.38a
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.
Problem 3.2.31a
31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).
a. For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.)
s(t)= −16t²+100t
Problem 3.1.23a
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 3x2 - 4x; P(1, -1)
Problem 3.8.42a
Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².
a. Find dr/dh for a cone with a lateral surface area of A=1500π.
Problem 3.8.60a
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
Problem 3.8.48a
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)
Problem 3.8.64.a
Vertical tangent lines
a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).
Problem 3.4.12a
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
Problem 3.5.72a
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
Problem 3.10.62a
62–65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x) = (x−1) sin^−1 x on [−1,1]
Problem 3.6.58a
{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?
Problem 3.43a
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?
Problem 3.1.25a
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)
Problem 3.6.50a
{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.
Problem 3.4.11a
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
Problem 3.8.14a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
x = e^y; (2, ln 2)
Problem 3.40a
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √3x; a= 12