Textbook Question
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
Ch. 3 - Derivatives
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 3, Problem 3.7.14
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x cos(2x + 3y) = y sin x
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Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use the chain rule.
Differentiate the left side: For x cos(2x + 3y), use the product rule. The derivative of x is 1, and the derivative of cos(2x + 3y) is -sin(2x + 3y) multiplied by the derivative of the inside function, which is 2 + 3(dy/dx).
Differentiate the right side: For y sin x, use the product rule again. The derivative of y is dy/dx, and the derivative of sin x is cos x.
Set the derivatives from both sides equal to each other. This will give you an equation involving dy/dx.
Solve the resulting equation for dy/dx to find the derivative of y with respect to x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. It involves differentiating both sides of an equation with respect to a variable, often x, while treating other variables, like y, as implicit functions of x. This method is essential when dealing with equations where y cannot be easily isolated.
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Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In implicit differentiation, the chain rule is often applied when differentiating terms involving y, as y is considered a function of x.
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Trigonometric Derivatives
Trigonometric derivatives are formulas used to find the derivatives of trigonometric functions such as sine, cosine, and tangent. For example, the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). In implicit differentiation problems involving trigonometric functions, these derivatives are crucial for correctly differentiating terms like x cos(2x + 3y) and y sin x.
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06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x³ - 3 (x² + π²)
1
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Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.
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Find the derivatives of all orders of the functions in Exercises 29–32.
y = x⁵ / 120
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = 1 x² csc 2
2 x