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Problem 3.2.33
Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2
Problem 3.9.27
Find the derivative of the following functions.
y = x² (1 - In x²)
Problem 3.8.28
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
y = xe^y
Problem 3.11.33
Piston compression A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s (see figure). What is the rate of change of the volume of the cylinder when the piston is 2 cm from the base of the chamber? <IMAGE>
Problem 3.10.59
Suppose the slope of the curve y=f^−1(x) at (4, 7) is 4/5. Find f′(7).
Problem 3.52
Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
h(x) = √x (√x-x³/²)
Problem 3.4.19
Derivatives Find and simplify the derivative of the following functions.
f(x) = 3x⁴(2x²−1)
Problem 3.8.91
90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.
x²(3y²−2y³) = 4
Problem 3.6.7
Define the acceleration of an object moving in a straight line.
Problem 3.5.57
Find y'' for the following functions.
y = x sin x
Problem 3.11.49
A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?
Problem 3.11.60
A ship leaves port traveling southwest at a rate of 12 mi/hr. At noon, the ship reaches its closest approach to a radar station, which is on the shore 1.5 mi from the port. If the ship maintains its speed and course, what is the rate of change of the tracking angle θ between the radar station and the ship at 1:30 P.M. (see figure)? (Hint: Use the Law of Sines.) <IMAGE>
Problem 3.9.32
Find the derivative of the following functions.
y = In(e^x + e^-x)
Problem 3.9.77
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (x+1)¹⁰ / (2x-4)⁸
Problem 3.5.2
How is lim x🠂0 sin x/x used in this section?
Problem 3.9.70
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In 2x/(x² + 1)³
Problem 3.19
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
y = x⁵
Problem 3.4.51
Derivatives Find and simplify the derivative of the following functions.
h(w) = w⁵/³ / w⁵/³+1
Problem 3.9.92
Calculate the derivative of the following functions (i) using the fact that bx = exIn b and (ii) using logarithmic differentiation. Verify that both answers are the same.
y = (x²+1)x
Problem 3.10.51
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=tan x; (1,π/4)
Problem 3.10.35
Evaluate the derivative of the following functions.
f(x) = sec-1 (ln x)
Problem 3.11.56
Two boats leave a port at the same time, one traveling west at 20 mi/hr and the other traveling southwest ( 45° south of west) at 15 mi/hr. After 30 minutes, how far apart are the boats and at what rate is the distance between them changing? (Hint: Use the Law of Cosines.)
Problem 3.11.31
A water heater that has the shape of a right cylindrical tank with a radius of 1 ft and a height of 4 ft is being drained. How fast is water draining out of the tank (in ft³/min) if the water level is dropping at 6 min/in?
Problem 3.5.23
Find the derivative of the following functions.
y = sin x + cos x
Problem 3.8.33
Use implicit differentiation to find dy/dx.
cos y2 + x = ey
Problem 3.5.21
Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.
Problem 3.1.58
Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim x🠂2) 1/x+1 - 1/3 / x-2
Problem 3.25
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
h(t) = t²/2 + 1
Problem 3.9.36
15–48. Derivatives Find the derivative of the following functions.
y = In (x³+1)^π
Problem 3.5.64
Find y'' for the following functions.
y = cos θ sin θ