Question: State the differentiation formula for each of

> 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.

> Describe an application of the differential equation y’ = ky(M - y).

> Describe an application of the differential equation y’ = k(M - y).

> Define the elasticity of demand, E(p), for a demand function. How is E(p) used?

> What is the difference between a relative rate of change and a percentage rate of change?

> State the formula for each of the following quantities: (a) The compound amount of P dollars in t years at interest rate r, compounded continuously (b) The present value of A dollars in n years at interest rate r, compounded continuously

> Explain how radiocarbon dating works.

> Given f (1) = 1, f ‘(1) = 5, g (1) = 3, g’(1) = 4, f ‘(3) = 2, and g ‘(3) = 6, compute. d/dx [ f (g (x))] |x=1

> What is meant by the half-life of a radioactive element?

> What is a growth constant? A decay constant?

> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’(2) = 1/3, compute the derivative: d/dx [x(g (x) - f (x))] |x=2

> What differential equation is key to solving exponential growth and decay problems? State a result about the solution to this differential equation.

> What is a logarithm?

> What are the coordinates of the reflection of the point (a, b) across the line y = x?

> State the properties that graphs of the form y = ekx have in common when k is positive and when k is negative.

> Write the differential equation satisfied by y = Cekt.

> What is e?

> State as many laws of exponents as you can recall.

> Differentiate the function using one or more of the differentiation rules discussed thus far. y = 2(2x - 1)5/4 (2x + 1)3/4

> Give an example of the use of logarithmic differentiation.

> State the four algebraic properties of the natural logarithm function.

> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’(2) = 1/3, compute the derivative: d/dx [x f (x)] |x=2

> Give the formula that converts a function of the form bx to an exponential function with base e.

> What is the difference between a natural logarithm and a common logarithm?

> State the two key equations giving the relationships between ex and ln x.

> State the main features of the graph of y = ln x.

> What is the x-intercept of the graph of the natural logarithm function?

> Outline the procedure for solving a related-rates problem.

> Differentiate the function using one or more of the differentiation rules discussed thus far. y = 2(x3 - 1)(3x2 + 1)4

> State the formula for d/dx yr, where y is defined implicitly as a function of x.

> Simplify the following expressions. ln 4 + ln 6 - ln 12

> Simplify the following expressions. 3 ln ½ + ln 16

> Simplify the following expressions. ½ ln 9

> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’(2) = 1/3, compute the derivative: d/dx [(g (x))2] |x=2

> Differentiate the functions. y = x√x

> Simplify the following expressions. ln x5 - ln x3

> Simplify the following expressions. ln 5 + ln x

> Find values of k and r for which the graph of y = kxr passes through the points (2, 3) and (4, 15).

> Determine the values of h and k for which the graph of y = hekx passes through the points (1, 6) and (4, 48).

> Differentiate the function using one or more of the differentiation rules discussed thus far. y = 5x3(2 - x)4

> In the study of epidemics, we find the equation ln(1 - y) - ln y = C - rt, where y is the fraction of the population that has a specific disease at time t. Solve the equation for y in terms of t and the constants C and r.

> Substantial empirical data show that, if x and y measure the sizes of two organs of a particular animal, then x and y are related by an allometric equation of the form ln y - k ln x = ln c, where k and c are positive constants that depend only on the typ

> Use logarithmic differentiation to differentiate the following functions. f (x) = x√x

> Use logarithmic differentiation to differentiate the following functions. f (x) = xx

> Use logarithmic differentiation to differentiate the following functions. f (x) = x√3

> Use logarithmic differentiation to differentiate the following functions. f (x) = 2x

> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’(2) = 1/3, compute the derivative: d/dx [(f (x))2] |x=2

> Use logarithmic differentiation to differentiate the following functions. f (x) = (x - 2)3(x - 3)4 / (x + 4)5

> Use logarithmic differentiation to differentiate the following functions. f (x) = (x + 1)(2x + 1)(3x + 1) / √(4x + 1)

> Use logarithmic differentiation to differentiate the following functions. f (x) = ex(3x - 4)8

> Differentiate the function using one or more of the differentiation rules discussed thus far. y = 6x2(x - 1)3

> The world’s population was 5.51 billion on January 1, 1993, and 5.88 billion on January 1, 1998. Assume that, at any time, the population grows at a rate proportional to the population at that time. In what year will the world’s population reach 7 billio

> Use logarithmic differentiation to differentiate the following functions. f (x) = (x + 1)4 (4x - 1)2

> Differentiate. y = ( ln 4x)( ln 2x)

> Differentiate. y = ln(3x + 1) ln(5x + 1)

> Differentiate. y = ln [(x + 1)4(x3 + 2)] / (x – 1)

> Differentiate. y = ln (x + 1)4 / ex-1

> Differentiate. y = ln (x + 1) / (x – 1)

> Differentiate. y = In [√xer²¹+1]

> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’(2) = 1/3, compute the derivative: d/dx [f (x)/g (x)] |x=2

> Differentiate. y = ln[e2x(x3 + 1)(x4 + 5x)]

> Differentiate. y = ln[(1 + x)2(2 + x)3(3 + x)4]

> Differentiate the function using one or more of the differentiation rules discussed thus far. y = (x4 + x2)10

> Differentiate. y = ln[(x + 1)(2x + 1)(3x + 1)]

> Differentiate. y = ln[(x + 5)(2x - 1)(4 - x)]

> Solve the given equation for x. ln[(x - 3)(x + 2)] - ln(x + 2)2 - ln 7 = 0

> Solve the given equation for x. ln(x + 1) - ln(x - 2) = 1

> Solve the given equation for x. 2( ln x)2 + ln x - 1 = 0

> Solve the given equation for x. ln √x = √ln x

> Solve the given equation for x. 3 ln x - ln 3x = 0

> Solve the given equation for x. ( ln x)2 - 1 = 0

> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’(2) = 1/3, compute the derivative: d/dx [ f (x)g (x)] |x=2