Back
Problem 3.3.6
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x³/3 + x²/2 + x/4
Problem 3.8.3
Assume that y = 5x and dx/dt = 2. Find dy/dt
Problem 3.2.43
In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.
f(x) = { 2x + tan x, x ≥ 0
x², x < 0
Problem 3.35
Find the derivatives of the functions in Exercises 1–42.
𝓻 = ( sin θ )²
( cos θ - 1 )
Problem 3.6.54
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
Problem 3.6.6
Derivative Calculations
In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).
y = sin u, u = x − cos x
Problem 3.6.58
In Exercises 41–58, find dy/dt.
y = √(3t + (√2 + √(1 − t)))
Problem 3.5.31
Derivatives
In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
Problem 3.7.39
Slopes, Tangent Lines, and Normal Lines
In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.
y = 2 sin(πx – y), (1,0)
Problem 3.6.36
Find the derivatives of the functions in Exercises 19–40.
g(t) = (1 + sin(3t) / (3 − 2t))⁻¹
Problem 3.1.10
In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
y = (1 / x³), (−2, −1/8)
Problem 3.4.11
Finding g on a small airless planet Explorers on a small airless planet used a spring gun to launch a ball bearing vertically upward from the surface at a launch velocity of 15 m/sec. Because the acceleration of gravity at the planet’s surface was gₛ m/sec², the explorers expected the ball bearing to reach a height of s = 15t − (1/2)gₛt² m t sec later. The ball bearing reached its maximum height 20 sec after being launched. What was the value of gₛ?
Problem 3.5.17
Derivatives
In Exercises 1–18, find dy/dx.
f(x) = x³ sin x cos x
Problem 3.9.32
Approximation Error
In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find
a. the change Δf = f(x₀ + dx) − f(x₀);
b. the value of the estimate df = fʹ(x₀) dx; and
c. the approximation error |Δf − df|.
f(x) = x⁴, x₀ = 1, dx = 0.1
Problem 3
Finding Derivative Functions and Values
Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.
g(t) = 1/t²; g′(−1), g′(2), g′(√3)
Problem 3.8.8
If x²y³ = 4/27 and dy/dt = ¹/₂, then what is dx/dt when x = 2?
Problem 3.8.5
If y = x² and dx/dt = 3, then what is dy/dt when x = –1?
Problem 3.1.30
Rates of Change
Speed of a rocket At t sec after liftoff, the height of a rocket is 3t² ft. How fast is the rocket climbing 10 sec after liftoff?
Problem 3.1.28
Find an equation of the straight line having slope 1/4 that is tangent to the curve y = √x.
Problem 3.2.44
In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.
g(x) = { 2x − x³ − 1, x ≥ 0
x − (1 / (x + 1)), x < 0
Problem 3.6.60
Second Derivatives
Find y'' in Exercises 59–64.
y = (1 − √x)⁻¹
Problem 3.3
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x³ - 3 (x² + π²)
Problem 3.6.78
Find dy/dt when x = 1 if y = x² + 7x − 5 and dx/dt = ¹/₃.
Problem 3.2.25
Using the Alternative Formula for Derivatives
Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.
g(x) = x / (x − 1)
Problem 3.6.41
In Exercises 41–58, find dy/dt.
y = sin²(πt − 2)
Problem 3.7.12
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x⁴ + sin y = x³y²
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 3.6.45
In Exercises 41–58, find dy/dt.
y = (t tan(t))¹⁰
Problem 3.2.41
In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.
f(x) = { 2x − 1, x ≥ 0
x² + 2x + 7, x < 0
Problem 3.6.77
Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.
