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Problem 3.6.46
In Exercises 41–58, find dy/dt.
y = (t⁻³/⁴ sin(t))⁴/³
Problem 3.6.51
In Exercises 41–58, find dy/dt.
y = (1 + tan⁴(t/12))³
Problem 3.1.18
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
f(x) = √(x + 1), (8, 3)
Problem 3.7.16
Find dr/dθ in Exercises 15–18.
r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴
Problem 3.1.33
Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).
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 3.17
Find the derivatives of the functions in Exercises 1–42.
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𝓻 = √2θ sinθ
Problem 3.3.55
For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.
lim (x → 1) (x⁵⁰ − 1) / (x − 1)
Problem 3.7.5
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x²(x – y)² = x² – y²
Problem 3.9.23
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sin(5√x)
Problem 3.3.59
The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).
Problem 3.7.23
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.
2√y = x – y
Problem 3.86
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.
(y - x)² = 2x + 4, (6, 2)
Problem 3.6.74
If r = sin(f(t)), f(0) = π/3, and f'(0) = 4, then what is dr/dt at t = 0?
Problem 3.7.14
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x cos(2x + 3y) = y sin x
Problem 3.6.54
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
Problem 3.9.33
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) = x⁻¹, x₀ = 0.5, dx = 0.1
Problem 3.5.31
Derivatives
In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
Problem 3.5.17
Derivatives
In Exercises 1–18, find dy/dx.
f(x) = x³ sin x cos x
Problem 3.31
Find the derivatives of the functions in Exercises 1–42.
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𝔂 = ( √ x )²
( 1 + x )
Problem 3.9.47
Tolerance The height and radius of a right circular cylinder are equal, so the cylinder’s volume is V = πh³. The volume is to be calculated with an error of no more than 1% of the true value. Find approximately the greatest error that can be tolerated in the measurement of h, expressed as a percentage of h.
Problem 3.1.14
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
g(x) = 8 / x², (2, 2)
Problem 3.2.42
In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.
g(x) = { x²/³, x ≥ 0
x¹/³, x < 0
Problem 3.3.44
Find the tangent line to the Witch of Agnesi (graphed here) at the point (2,1).
Problem 3.9.13
Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.
Problem 3.35
Find the derivatives of the functions in Exercises 1–42.
𝓻 = ( sin θ )²
( cos θ - 1 )
Problem 3.67
a. Graph the function
ƒ(x) = { x², -1 ≤ x < 0
{ -x², 0 ≤ x ≤ 1.
b. Is ƒ continuous at x = 0?
c. Is ƒ differentiable at x = 0?
Give reasons for your answers.
Problem 3.7.45
The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y⁴ – 4y² = x⁴ – 9x² at the four indicated points.
Problem 3.47
In Exercises 43–50, find by implicit differentiation.
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√xy = 1
Problem 3.9.27
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = 3 csc(1 − 2√x)
