The body mass index, or BMI, is a ratio of a person’s weight divided by the square of his or her height. Let b (t) denote the BMI; then, b (t) = w(t) / [h(t)]2, where t is the age of the person, w(t) the weight in kilograms, and h(t) the height in meters. Find an expression for b’ (t).
> Use logarithmic differentiation to differentiate the function. f (x) = [(xex)/(x3 + 3)]
> Use logarithmic differentiation to differentiate the function. f (x) = √(x2 + 5) ex2
> Use logarithmic differentiation to differentiate the function. f (x) = 10x
> Use logarithmic differentiation to differentiate the function. f (x) = x1+x
> Use logarithmic differentiation to differentiate the function. f (x) = (x2 + 5)6(x3 + 7)8(x4 + 9)10
> Use logarithmic differentiation to differentiate the function. f (x) = bx, where b > 0
> Use logarithmic differentiation to differentiate the function. f (x) = x√x
> Function h(x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = f (x2)/x
> Use logarithmic differentiation to differentiate the function. f (x) = 2x
> Use logarithmic differentiation to differentiate the function. f (x) = 5√(x5 + 1)/(x5 + 5x + 1)
> Differentiate the function. y = ln(ex + 3e-x)
> Write expression in the form 2kx or 3kx, for a suitable constant k. 4x, (√3)x, (1/9)x
> Differentiate the functions. y = (x2 + 3)(x2 - 3)10
> Differentiate the function. y = ln (1 /e√x)
> Differentiate the function. y = e2 ln(2x+1)
> Differentiate the function. y = ln |x – 1|
> Differentiate the function. y = ln(3x+1) - ln 3
> Differentiate the function. y = ln(2x)
> Function h(x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = f (f (x))
> Differentiate the function. y = ln 3√(x3 + 3x – 2)
> Differentiate the function. y = ln(ex2/x)
> Differentiate the function. y = ln |-2x + 1|
> Differentiate the function. y = ln √[(x2 + 1) / (2x + 3)]
> Differentiate the function. y = ln(x2 + ex)
> The relationship between the area of the pupil of the eye and the intensity of light was analyzed by B. H. Crawford. Crawford concluded that the area of the pupil is square millimeters when x units of light are entering the eye per unit time. (Source:
> Differentiate the function. y = ex ln x
> Differentiate the function. y = 1 / ln x
> Differentiate the function. y = ln( ln√x)
> Differentiate the function. y = e2 ln (x+1)
> Function h(x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = -f (-x)
> Differentiate the function. y = x ln x - x
> Differentiate the function. y = ln[e6x(x2 + 3)5(x3 + 1)-4]
> Differentiate the function. y = ln [xex / √(1 + x)]
> Differentiate the function. y = (x ln x)3
> Differentiate the function. y = ( ln x)2
> Differentiate the function. y = ln(9x)
> The BMI is usually used as a guideline to determine whether a person is overweight or underweight. For example, according to the Centers for Disease Control, a 12-year-old boy is at risk of being overweight if his BMI is between 21 and 24 and is consider
> Differentiate the function. y = ln (5x - 7)
> Suppose that a kitchen appliance company’s monthly sales and advertising expenses are approximately related by the equation xy - 6x + 20y = 0, where x is thousands of dollars spent on advertising and y is thousands of dishwashers sold. Currently, the com
> Animal physiologists have determined experimentally that the weight W (in kilograms) and the surface area S (in square meters) of a typical horse are related by the empirical equation S = 0.1W2/3. How fast is the surface area of a horse increasing at a t
> Function h(x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = 2 f (2x + 1)
> At the beginning of 1990, 20.2 million people lived in the metropolitan area of Mexico City, and the population was growing exponentially. The 1995 population was 23 million. (Part of the growth is due to immigration.) If this trend continues, how large
> An offshore oil well is leaking oil onto the ocean surface, forming a circular oil slick about .005 meter thick. If the radius of the slick is r meters, the volume of oil spilled is V = .005πr2 cubic meters. If the oil is leaking at a constant rate of 20
> Suppose that the price p and quantity x of a certain commodity satisfy the demand equation 6p + 5x + xp = 50 and that p and x are functions of time, t. Determine the rate at which the quantity x is changing when x = 4, p = 3, and dp/dt = -2.
> A town library estimates that, when the population is x thousand persons, approximately y thousand books will be checked out of the library during 1 year, where x and y are related by the equation y3 - 8000x2 = 0. (a) Use implicit differentiation to find
> A factory’s weekly production costs y and its weekly production quantity x are related by the equation y2 - 5x3 = 4, where y is in thousands of dollars and x is in thousands of units of output. (a) Use implicit differentiation to find a formula for dy/dx
> x and y are related by the given equation. Use implicit differentiation to calculate the value of dy/dx for the given values of x and y. xy2 - x3 = 10; x = 2, y = 3
> x and y are related by the given equation. Use implicit differentiation to calculate the value of dy/dx for the given values of x and y. x2 - xy3 = 20; x = 5, y = 1
> x and y are related by the given equation. Use implicit differentiation to calculate the value of dy/dx for the given values of x and y. xy4 = 48; x = 3, y = 2
> x and y are related by the given equation. Use implicit differentiation to calculate the value of dy/dx for the given values of x and y. x2y2 = 9; x = 1, y = 3
> The graph of x3 + y3 = 9xy is the folium of Descartes shown in Fig. 4. (a) Find dy/dx by implicit differentiation. (b) Find the slope of the curve at (2, 4). Figure 4: x+y+1=0 Asymptote Y x x³ + y² = 9xy g
> Differentiate the function. y = x2 / (x2 + 1)2
> The graph of x2/3 + y2/3 = 8 is the astroid in Fig. 3. (a) Find dy/dx by implicit differentiation. (b) Find the slope of the tangent line at (8, -8). Figure 3: ม 22/3 + 2/3 = 8
> The amount, A, of anesthetics that a certain hospital uses each week is a function of the number, S, of surgical operations performed each week. Also, S, in turn, is a function of the population, P, of the area served by the hospital, while P is a functi
> The revenue, R, that a company receives is a function of the weekly sales, x. Also, the sales level, x, is a function of the weekly advertising expenditures, A, and A, in turn, is a varying function of time. (a) Write the derivative symbols for the follo
> Refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h ‘(1). h(x) = g( f (x)) Figure 2: 4 3 2 0 2 y=f(x) 0 2 3 W y = g(x) 2 co 3
> Refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h ‘(1). h(x) = f (g (x)) Figure 2: 4 3 2 0 2 y=f(x) 0 2 3 W y = g(x) 2 co 3
> Refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h ‘(1). h(x) = [f (x)]2 Figure 2: 4 3 2 0 2 y=f(x) 0 2 3 W y = g(x) 2 co 3
> Refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h ‘(1). h(x) = f (x)/g (x) Figure 2: 4 3 2 0 2 y=f(x) 0 2 3 W y = g(x) 2 co 3
> Refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h ‘(1). h(x) = f (x) * g (x) Figure 2: 4 3 2 0 2 y=f(x) 0 2 3 W y = g(x) 2 co 3
> Apply the special case of the general power rule d/dx [h(x)]2 = 2h(x)h ’(x) and the identity fg = ¼ [(f + g)2 - (f - g)2] to prove the product rule.
> Refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h ‘(1). h(x) = 2f (x) - 3g (x) Figure 2: 4 3 2 0 2 y=f(x) 0 2 3 W y = g(x) 2 co 3
> Function h(x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = f (x2)
> Find dy/dx, where y is a function of u such that dy/du = u/√(1 + u4). u = 2/x
> Find dy/dx, where y is a function of u such that dy/du = u/√(1 + u4). u = √x
> Find dy/dx, where y is a function of u such that dy/du = u/√(1 + u4). u = x2
> Find dy/dx, where y is a function of u such that dy/du = u/(u2 + 1). State the answer in terms of x only. u = 5/x
> Find dy/dx, where y is a function of u such that dy/du = u/(u2 + 1). State the answer in terms of x only. u = x2 + 1
> Find dy/dx, where y is a function of u such that dy/du = u/(u2 + 1). State the answer in terms of x only. u = x3/2
> Find a formula for d/dx f (g (x)), where f (x) is a function such that f (x) = x√(1 - x2). g (x) = x3/2
> Find a formula for d/dx f (g (x)), where f (x) is a function such that f (x) = x√(1 - x2). g (x) = √x
> Find a formula for d/dx f (g (x)), where f (x) is a function such that f (x) = x√(1 - x2). g (x) = x2
> Let f (x), g (x), and h (x) be differentiable functions. Find a formula for the derivative of f (x)g (x)h (x).
> Given f (1) = 1, f ‘(1) = 5, g (1) = 3, g ’(1) = 4, f ‘(3) = 2, and g ‘(3) = 6, compute. d/dx [ g ( g (x))] |x=1
> Find a formula for d/dx f (g (x)), where f (x) is a function such that f (x) = 1/(x2 + 1). g (x) = x2 + 1
> Find a formula for d/dx f (g (x)), where f (x) is a function such that f (x) = 1/(x2 + 1). g (x) = 1/x
> Find a formula for d/dx f (g (x)), where f (x) is a function such that f (x) = 1/(x2 + 1). g (x) = x3
> A company pays y dollars in taxes when its annual profit is P dollars. If y is some (differentiable) function of P and P is some function of time t, give a chain rule formula for the time rate of change of taxes dy/dt.
> A store estimates that its cost when selling x lamps per day is C dollars, where C = 40x + 30 (the marginal cost per lamp is $40). If daily sales are rising at the rate of three lamps per day, how fast are the costs rising? Explain your answer using the
> Repeat Exercise 17, with the sidewalk on the inside of all four sides. In this case, the 800-square-meter planted region has dimensions x - 4 meters by y - 4 meters. Exercise 17: A botanical display is to be constructed as a rectangular region with a ri
> A botanical display is to be constructed as a rectangular region with a river as one side and a sidewalk 2 meters wide along the inside edges of the other three sides. (See Fig. 1.) The area for the plants must be 800 square meters. Find the outside dime
> Find the equation of the line tangent to the graph of y = x – 3/√(4 + x2) at the point where x = 0.
> Find the equation of the line tangent to the graph of y = (x3 - 1)(x2 + 1)4 at the point where x = -1.
> Let f (x) = (x2 + 1)/(x2 + 5). Find all x such that f ‘ (x) = 0.
> Given f (1) = 1, f ‘(1) = 5, g (1) = 3, g ’(1) = 4, f ‘(3) = 2, and g ‘(3) = 6, compute. d/dx [ f ( f (x))] |x=1
> The derivative of (x3 - 4x)>x is obviously 2x for x ≠ 0, because (x3 - 4x)/x = x2 ≠ 4 for x 0. Verify that the quotient rule gives the same derivative.
> Let f (x) = (3x + 1)4(3 - x)5. Find all x such that f ‘(x) = 0.
> Differentiate the following functions. y = (x3 + x)/(x2 – x)
> Differentiate the following functions. y = [(3 - x2)/x3]2
> Differentiate the following functions. y = 2x/(2 - 3x)
> Differentiate the following functions. y = (x2 - 6x)/(x – 2)
> Differentiate the following functions. y = 1/(x2 + 5x + 1)6
> Differentiate the following functions. y = 3(x2 - 1)3(x2 + 1)5
> Differentiate the following functions. y = √x/(√x + 4)
> What does it mean for a function to be defined implicitly by an equation?
> Given f (1) = 1, f ‘(1) = 5, g (1) = 3, g ’(1) = 4, f ‘(3) = 2, and g ‘(3) = 6, compute. d/dx [ g ( f (x))] |x=1
> What is the relationship between the chain rule and the general power rule?
> Let f (x) = 1/x and g (x) = x3. (a) Show that the product rule yields the correct derivative of (1/x)x3 = x2. (b) Compute the product f (x)g (x), and note that it is not the derivative of f (x)g (x).
> State the chain rule. Give an example.
> State the product rule and the quotient rule.
