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Problem 3.6.5
Suppose w(t) is the weight (in pounds) of a golden retriever puppy t weeks after it is born. Interpret the meaning of w'(15) = 1.75.
Problem 3.9.27
Find the derivative of the following functions.
y = x² (1 - In x²)
Problem 3.9.40
15–48. Derivatives Find the derivative of the following functions.
y = 4^-x sin x
Problem 3.4.21
Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1
Problem 3.10.71
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = e^3x+1
Problem 3.10.9
If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.
Problem 3.8.5
5–8. Calculate dy/dx using implicit differentiation.
x = y²
Problem 3.9.46
15–48. Derivatives Find the derivative of the following functions.
y = 10^x(In 10^x-1)
Problem 3.11.21
A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)
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 3.10.4
How are the derivatives of sin^−1 x and cos^−1 x related?
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 3.45
Derivative calculations Evaluate the derivative of the following functions at the given point.
f(s) = 2√s-1; a=25
Problem 3.5.89
{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = f(x+0.01)-f(x) / 0.01 For the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?
f(x) = sin x on [−π,π]
Problem 3.9.16
Find the derivative of the following functions.
y = x² In x
Problem 3.11.17
A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?
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 3.5.17
Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (tan 7x) / (sin x)
Problem 3.8.35
Use implicit differentiation to find dy/dx.
x3 = (x + y) / (x - y)
Problem 3.9.44
15–48. Derivatives Find the derivative of the following functions.
P = 40/1+2^-t
Problem 3.9.36
15–48. Derivatives Find the derivative of the following functions.
y = In (x³+1)^π
Problem 3.9.65
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).
y = (cos x) In cos²x
Problem 3.4.23
Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t
Problem 3.2.76
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
Graph the following curves and determine the location of any vertical tangent lines.
a. x²+y² = 9
Problem 3.10.44
Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sec−1(ex); (ln 2,π/3)
Problem 3.9.7
Simplify the expression e^xln(x²+1).
Problem 3.5.29
Find the derivative of the following functions.
y = cos x/sin x + 1
Problem 3.11.16
The edges of a cube increase at a rate of 2 cm/s. How fast is the volume changing when the length of each edge is 50 cm?
Problem 3.5.11
Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (sin 3x) / x
Problem 3.1.14
A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s (t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a.
s(t) = -16t2 + 128t + 192; a = 2