Textbook Question
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.6.47
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
47. y=(arccot(x³))³
Verified step by step guidance1
Identify the function y = (arccot(x^3))^3 as a composition of functions, where the outer function is u^3 and the inner function is u = arccot(x^3).
Apply the chain rule for differentiation: \( \frac{dy}{dx} = 3 (arccot(x^3))^2 \cdot \frac{d}{dx}[arccot(x^3)] \).
Recall the derivative of \( arccot(z) \) with respect to z is \( -\frac{1}{1+z^2} \). Here, \( z = x^3 \), so use the chain rule again to find \( \frac{d}{dx}[arccot(x^3)] = -\frac{1}{1+(x^3)^2} \cdot \frac{d}{dx}[x^3] \).
Calculate \( \frac{d}{dx}[x^3] = 3x^2 \), then substitute back to get \( \frac{d}{dx}[arccot(x^3)] = -\frac{3x^2}{1+x^6} \).
Combine all parts to write the derivative as \( \frac{dy}{dx} = 3 (arccot(x^3))^2 \cdot \left(-\frac{3x^2}{1+x^6}\right) \), which simplifies the expression for the derivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chain Rule
The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
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Intro to the Chain Rule
Derivative of Inverse Trigonometric Functions
Inverse trigonometric functions like arccot(x) have specific derivative formulas. The derivative of arccot(x) with respect to x is -1/(1 + x²). Knowing these derivatives is essential when differentiating expressions involving inverse trig functions.
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06:35Derivatives of Other Inverse Trigonometric Functions
Power Rule
The power rule is used to differentiate functions of the form y = [u(x)]^n, where n is a constant. It states that the derivative is n times the function raised to the (n-1) power multiplied by the derivative of the inner function u(x). This rule is often combined with the chain rule for composite functions.
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5:50Power Rules
Related Practice
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Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
9. (sinh(x)+cosh(x))⁴
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130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?
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90. Find f'(0) for
f(x) = e^(-1/x²), x≠0
= 0, x = 0.
1
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In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))
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In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)