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Problem 3.13
Find the derivatives of the functions in Exercises 1–42.
s = cos⁴ (1 - 2t)
Problem 3.21
Find the derivatives of the functions in Exercises 1–42.
𝔂 = 1 x² csc 2
2 x
Problem 3.6.5
Derivative Calculations
In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).
y = √u, u = sin x
Problem 3.2.39
One-Sided Derivatives
Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.
Problem 3.7.12
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x⁴ + sin y = x³y²
Problem 3.9.4
Finding Linearizations
In Exercises 1–5, find the linearization L(x) of f(x) at x = a.
f(x) = ∛x, a = −8
Problem 3.7.43
The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.
Problem 3.9.22
Derivatives in Differential Form
In Exercises 17–28, find dy.
xy² − 4x³/² − y = 0
Problem 3.7.20
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
x²/³ + y²/³ = 1
Problem 3.7.52
Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?
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Problem 3.8.33
A balloon and a bicycle A balloon is rising vertically above a level, straight road at a constant rate of 1 ft/sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/sec passes under it. How fast is the distance s(t) between the bicycle and the balloon increasing 3 sec later?
Problem 3.7.47
Theory and Examples
Intersecting normal line The line that is normal to the curve x² + 2xy – 3y² = 0 at (1,1) intersects the curve at what other point?
Problem 3.33
Find the derivatives of the functions in Exercises 1–42.
_____
𝔂 = / x² + x
√ x²
Problem 3.6.89
Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².
Problem 3.3.9
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = 6x² − 10x − 5x⁻²
Problem 3.7.25
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
3 + sin y = y – x³
Problem 3.6.72
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1
Problem 3.88
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
x³/² + 2y³/² = 17, (1, 4)
Problem 3.9.23
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sin(5√x)
Problem 3.6.69
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = cot(πu/10), u = g(x) = 5√x, x = 1
Problem 3.6.45
In Exercises 41–58, find dy/dt.
y = (t tan(t))¹⁰
Problem 3.5.15
Derivatives
In Exercises 1–18, find dy/dx.
y = (sec x + tan x)(sec x − tan x)
Problem 3.3.34
Find the first and second derivatives of the functions in Exercises 33–38.
s = (t² + 5t − 1) / t²
Problem 3.5.52
Is there a value of b that will make
g(x) = { x + b, x < 0
cos x, x ≥ 0
continuous at x = 0? Differentiable at x = 0? Give reasons for your answers.
Problem 3.7.49
Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?
Problem 3.6.21
Find the derivatives of the functions in Exercises 19–40.
s = (4 / 3π)sin(3t) + (4 / 5π)cos(5t)
Problem 3.7.22
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
y² – 2x = 1 – 2y
Problem 3.57
Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.
x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4
Find the first derivatives of the following combinations at the given value of x.
a. 6ƒ(x) - g(x), x = 1
Problem 3.6.77
Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.
Problem 3.9.20
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = (2√x)/(3(1 + √x))
