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mathematics
applied calculus
Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions
State the chain rule. Give an example.
Compute f(g(x)), where f(x) and g(x) are the following: f(x) = x + 1 X-3' g(x) = x + 3
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3 + y3 - 6 = 0
What is the relationship between the chain rule and the general power rule?
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.y5 - 3x2 = x
Differentiate the functions in Exercises. y = x(x² + 1)4
Differentiate the functions in Exercises. y=xVx
What does it mean for a function to be defined implicitly by an equation?
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x4 + (y + 3)4 = x2
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.y4 - x4 = y2 - x2
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x).h(x) = (x3 + 8x - 2)5
Outline the procedure for solving a related-rates problem.
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x). h(x) = V4-x²
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3 + y3 = x2 + y2
Differentiate the functions in Exercises. y = (x² + 3)(x² - 3)10
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x).h(x) = (9x2 + 2x - 5)7
Differentiate the functions in Exercises. y=[(-2x³ + x)(6x - 3)]4
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x). h(x) = 1 r3 – 5,2+1
Differentiate the functions in Exercises. y = x²(3x4 + 12x - 1)²
Differentiate the functions in Exercises. y=(5r+1)(x2 –1)+ 2x + 1 3
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.2x3 + y = 2y3 + x
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x). h(x) = (4x - 3)³ + 1 4x 3 -
Differentiate the functions in Exercises. y= x-1 x + 1
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x4 + 4y = x - 4y3
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x).h(x) = (5x2 + 1)-1/2
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.xy = 5
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.xy3 = 2
Differentiate the functions in Exercises. y 1 x² + x + 7 2
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = (x2 + 5)15
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x(y + 2)5 = 8
Differentiate the functions in Exercises. y = - 1 2 x² + 1 X
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = (x4 + x2)10
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x2y3 = 6
Differentiate the functions in Exercises. y X x +
Let f(x) = (3x + 1)4(3 - x)5. Find all x such that f′(x) = 0.
Differentiate the functions in Exercises. y x + 3 (2x + 1)²
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = 6x2(x - 1)3
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3y2 - 4x2 = 1
Let f(x) = (x2 + 1)/(x2 + 5). Find all x such that f′(x) = 0.
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = 5x3(2 - x)4
Find the equation of the line tangent to the graph ofat the point where x = 0. y= x-3 √4+x²
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.(x + 1)2 (y - 1)2 = 1
Find the equation of the line tangent to the graph of y = (x3 - 1)(x2 + 1)4 at the point where x = -1.
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 dimensions of the region that minimizes the area
Differentiate the functions in Exercises. y || 1 | TT + 2 x² + 1
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = 2(x3 - 1)(3x2 + 1)4
Given f(1) = 1, f′(1) = 5, g(1) = 3, g′(1) = 4, f′(3) = 2, and g′(3) = 6, compute the following derivatives: d dx [f(g(x)) g(x))] x=1
Differentiate the functions in Exercises. 4 y = x² +4√x
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3 + y3 = x3y3
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = 2(2x - 1)5/4(2x + 1)3/4
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 17A botanical display is to be constructed as a rectangular region with a river as one side and a sidewalk 2 meters wide
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x2 + 4xy + 4y = 1
Given f(1) = 1, f′(1) = 5, g(1) = 3, g′(1) = 4, f′(3) = 2, and g′(3) = 6, compute the following derivatives: [=x ОЛА xp P
Given f(1) = 1, f′(1) = 5, g(1) = 3, g′(1) = 4, f′(3) = 2, and g′(3) = 6, compute the following derivatives: d | dx [g (f(x))] x=1
Differentiate the functions in Exercises. y 1-2 (x² + 1)²
Differentiate the functions in Exercises. y= ax + b cx + d
Given f(1) = 1, f′(1) = 5, g(1) = 3, g′(1) = 4, f′(3) = 2, and g′(3) = 6, compute the following derivatives: d dx [g(g(x))] x=1
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x2y + y2x = 3
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3y + xy3 = 4
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 chain rule.
Differentiate the functions in Exercises. (x + 1)³ 2 (x - 5)²
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.4y3 - x2 = -5; x = 3, y = 1
Differentiate the functions in Exercises. y = [(3x² + 2x + 2)(x - 2)]²
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.
Differentiate the functions in Exercises. y 1 V √x - 2
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point. xy³ = 2; x = − 1, y = −2
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.y2 = x3 + 1; x = 2, y = -3
Differentiate the functions in Exercises. y 1 √x + 1
In Exercises, 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
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = f(x2)
In Exercises, 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
Differentiate the functions in Exercises. y = 3 √x+1
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = 2 f(2x + 1)
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.√x + √y = 7; x = 9, y = 16
In Exercises, 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
Differentiate the functions in Exercises. y = 3 (x + 11) ³ - 3
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = -f (-x)
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x). h(x) = f(x²)
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.xy + y3 = 14; x = 3, y = 2
In Exercises, 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
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x). h(x) = √f(x²)
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = f(f (x))
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.y2 = 3xy - 5; x = 2, y = 1
In Exercises, 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 the equation of the tangent line to the graph of x2 y4 = 1 at the point (4, 1/2) and at the point (4, - 1/2).
In Exercises, 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
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 function of time, t.(a) Write the derivative
Differentiate the functions in Exercises. y = x + 3 2 x² + 1
Differentiate the functions in Exercises. y = √3x - 1 X
Differentiate the functions in Exercises. y = √x + 2(2x + 1)²
In Exercises, find dy/dx, where y is a function of u such thatState the answer in terms of x only.u = x2 + 1 dy U du +1 u²
Find the equation of the tangent line to the graph of x4 y2 = 144 at the point (2, 3) and at the point (2, -3).
Suppose that x and y represent the amounts of two basic inputs for a production process and that the equationdescribes all input amounts where the output of the process is 1080 units(a) Find dy/dx.(b) What is the marginal rate of substitution of x for y when x = 16 and y = 54? 30x¹/3,2/3 = 1080
In Exercises, find dy/dx, where y is a function of u such thatState the answer in terms of x only. dy U du +1 u²
Sketch the graph of y = 4x/(x + 1)2, x > -1.
In Exercises, find dy/dx, where y is a function of u such thatu = x2 dy du U V1 + u
Compute d/dx f(g (x)), where f(x) and g(x) are the following: f(x)=√x, g(x) = x² + 1
Sketch the graph of y = 2/(1 + x2).
In Exercises, find dy/dx, where y is a function of u such thatu = √x dy du U V1 + u
Compute d/dx f(g (x)), where f(x) and g(x) are the following:f(x) = x5, g (x) = 6x - 1
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