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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

In Exercises, find the derivative of each function. y -2 Vx
Find the derivative of each function. y 3 - 1 2 ln x
In Exercises, find the derivative of each function. f(x) = x³ + 5 X
In Exercises, use the quotient rule to find the derivative of each function. p(t) = Vt t - 1
Write each function as the composition of two functions. (There may be more than one way to do this.)y = ex2
Use the rules for derivatives to find the derivative of each function defined as follows. r(t = 5t² - 7t (3t+1)³
In Exercises, use the quotient rule to find the derivative of each function. r(t) = Vt 2t + 3
Use the rules for derivatives to find the derivative of each function defined as follows.y = 3x(2x + 1)3
Write each function as the composition of two functions. (There may be more than one way to do this.)y = ln(4x + 7)
In Exercises, find the derivative of each function. g(x) = x³ - 4x Vx
Find the derivative of each function. y = Vinx - 3|
Find the derivative of each function.y = (ln |x + 1|)4
In Exercises, use the quotient rule to find the derivative of each function. y = 9 + XS Vx
Use the rules for derivatives to find the derivative of each function defined as follows.y = 4x2(3x - 2)5
Write each function as the composition of two functions. (There may be more than one way to do this.)y = ln(1 + x3)
In Exercises, find the derivative of each function.y = (8x4 - 5x2 + 1)4
In Exercises, find the derivative of each function.g(x) = (8x2 - 4x)2
Find the derivative of each function.y = ln |ln x|
In Exercises, find the derivative of each function.k(x) = -2(12x2 + 5)-6
Which of the following describes the derivative function ƒ′(x) of a quadratic function ƒ(x)?(a) Quadratic (b) Linear(c) Constant (d) Cubic (third degree)
In Exercises, use the quotient rule to find the derivative of each function. g(y) = || 1.4 + 1 2.5 + 2
Find the derivative of each function.y = ex2 ln x
Use the rules for derivatives to find the derivative of each function defined as follows.g(t) = t3(t4 + 5)7/2
In Exercises, use the quotient rule to find the derivative of each function. f(x) (3x² + 1)(2x - 1) 5x+4
In Exercises, find the derivative of each function.ƒ(x) = -7(3x4 + 2)-4
Which of the following describes the derivative function ƒ′(x) of a cubic (third degree) function ƒ(x)?(a) Quadratic (b) Linear(c) Constant (d) Cubic
Find the derivative of each function.y = e2x-1 ln(2x - 1)
Use the rules for derivatives to find the derivative of each function defined as follows.y = -6e2x
In Exercises, find the derivative of each function.s(t) = 45(3t3 - 8)3/2
Explain the relationship between the slope and the derivative of ƒ(x) at x = a.
Find the derivative of each function. y e.t In x
In Exercises, use the quotient rule to find the derivative of each function. g(x) (2x² + 3)(5x + 2) 6x - 7
Which of the following do not equal(a)(b)(c)(d) 12x3 + 12x-3 d dx -(4x² - 6x-²)?
Find the derivative of each function. p(y) = In y ey
Use the rules for derivatives to find the derivative of each function defined as follows.y = 8e0.5x
Find the derivative of each function. s(t) = √e¹ + In 2t
In Exercises, find the derivative of each function.s(t) = 12(2t4 + 5)3/2
Find the error in the following work. 2x D. (²x + 5) = Dx 2 - (2x + 5)(2x) - (x² - 1)2 (x² - 1)² 4x² + 10x 2x² + 2 (x² - 1)² 2x² + 10x + 2 (x² - 1)²
Use the rules for derivatives to find the derivative of each function defined as follows.y = e-2x3
If g(3) = 4, g′(3) = 5, ƒ(3) = 9, and ƒ′(3) = 8, find h′(3) when h(x) = ƒ(x)g(x).
In Exercises, find the derivative of each function.g(t) = -3√7t3 - 1
Find the derivative of each function.g(z) = (e2z + ln z)3
Use the rules for derivatives to find the derivative of each function defined as follows.y = -4ex2
If g(3) = 4, g′(3) = 5, ƒ(3) = 9, and ƒ′(3) = 8, find h′(3) when h(x) = ƒ(x)/g(x).
In Exercises, find the derivative of each function.ƒ(t) = 8√4t2 + 7
Find the error in the following work. Dx - x³ 4 x³(2x) - (x² - 4)(3x²) = 2x4 - 3x² + 12x² = −x² + 12x² =
Find the derivative of each function.y = log(6x)
Use the rules for derivatives to find the derivative of each function defined as follows.y = 5xe2x
In Exercises, find the derivative of each function.m(t) = -6t(5t4 - 1)4
Find the derivative of each function.y = log(4x - 3)
Use the rules for derivatives to find the derivative of each function defined as follows.y = -7x2e-3x
In Exercises, find the derivative of each function.r(t) = 4t(2t5 + 3)4
Find the derivative of each function.y = log |1 - x|
Use the rules for derivatives to find the derivative of each function defined as follows.y = ln(2 + x2)
Use the rules for derivatives to find the derivative of each function defined as follows. y = In |3x| x - 3
In Exercises, find the equation of the tangent line to the graph of the given function at the given point.ƒ(x) = (2x2 - 4)(x + 2) at (2, 16)
Use the rules for derivatives to find the derivative of each function defined as follows. y In 2x 1 x + 3
In Exercises, find the derivative of each function.y = (x3 + 2)(x2 - 1)4
In Exercises, find the slope and the equation of the tangent line to the graph of the given function at the given value of x.y = -3x5 - 8x3 + 4x2; x = 1
Find the derivative of each function.y = log5√5x + 2
In Exercises, find the equation of the tangent line to the graph of the given function at the given point.ƒ(x) = √x/(4x - 2) at (1, 0.5)
In Exercises, find the derivative of each function.q(y) = 4y2(y2 + 1)5/4
In Exercises, find the slope and the equation of the tangent line to the graph of the given function at the given value of x.y = -2x1/2 + x3/2; x = 9
Find the derivative of each function.y = log7√4x - 3
In Exercises, find the equation of the tangent line to the graph of the given function at the given point.ƒ(x) = (x2 - 3)/(2x + 1) at (-1, 2)
In Exercises, find the derivative of each function.p(z) = z(6z + 1)4/3
In Exercises, find the derivative of each function. y || -5 (2x³ + 1)²
Use the rules for derivatives to find the derivative of each function defined as follows. y || xex In(x² - 1)
Use the rules for derivatives to find the derivative of each function defined as follows. y || (x² + 1)e²x In x
In Exercises, find the derivative of each function. y || 1 (3x² - 4)5
In Exercises, find the slope and the equation of the tangent line to the graph of the given function at the given value of x.y = -x-3 + x-2; x = 2
Find the derivative of each function.y = log3 (x2 + 2x)3/2
In Exercises, find the derivative of each function. r(t) = = (5t - 6)4 31² + 4
Find all points on the graph of ƒ(x) = 9x2 - 8x + 4 where the slope of the tangent line is 0.
Find the derivative of each function.y = log2 (2x2 - x)5/2
What is the result of applying the product rule to the function ƒ(x) = kg(x), where k is a constant? Compare with the rule for differentiating a constant times a function from the previous section.
Find all points on the graph of ƒ(x) = x3 + 9x2 + 19x - 10 where the slope of the tangent line is -5.
Find the derivative of each function.w = log8 (2p - 1)
Use the rules for derivatives to find the derivative of each function defined as follows.s = (t2 + et)2
Following the steps used to prove the product rule for derivatives, prove the quotient rule for derivatives.
In Exercises, find the derivative of each function. p(t) = (2t + 3)³ 41² 1
In Exercises, for each function find all values of x where the tangent line is horizontal.ƒ(x) = 2x3 + 9x2 - 60x + 4
In Exercises, find the derivative of each function. y || 3x² X (2x - 1)³
Find the derivative of each function.z = 10y log y
Use the rules for derivatives to find the derivative of each function defined as follows.q = (e2p+1 - 2)4
Use the fact that ƒ(x) = u(x)/v(x) can be rewritten as ƒ(x)v(x) = u(x) and the product rule for derivatives to verify the quotient rule for derivatives.
In Exercises, for each function find all values of x where the tangent line is horizontal.ƒ(x) = x3 + 15x2 + 63x - 10
Find the derivative of each function.ƒ(x) = e√x ln1(√x + 5)
Use the rules for derivatives to find the derivative of each function defined as follows.y = 3 · 10-x2
For each function, find the value(s) of x in which f′(x) = 0, to 3 decimal places.ƒ(x) = (x2 - 2)(x2 - √2)
In Exercises, for each function find all values of x where the tangent line is horizontal.ƒ(x) = x3 - 4x2 - 7x + 8
Find the derivative of each function.ƒ(x) = ln(xe√x + 2)
Use the rules for derivatives to find the derivative of each function defined as follows.y = 10 · 2√x
In Exercises, find the derivative of each function. y = x² + 4x (3x³ + 2)²
Find the derivative of each function. f(t) = In(t² + 1) + t In(t² + 1) + 1
For each function, find the value(s) of x in which f′(x) = 0, to 3 decimal places. f(x) = x - 2 X .2 x² + 4
In Exercises, for each function find all values of x where the tangent line is horizontal.ƒ(x) = x3 - 5x2 + 6x + 3
Find the derivative of each function. f(t) = 213/2 In (21³/2 + 1)
Gottfried Leibniz, one of the inventors of calculus, initially thought that the derivative of ƒ(x)g(x) was ƒ′(x)g′(x), based on the example ƒ(x) = x2 + bx and g(x) = cx + d. Besides neglecting
Use the rules for derivatives to find the derivative of each function defined as follows.h(z) = log(1 + ez)

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