Textbook Question
Find the slope of the curve x³y³ + y² = x + y at the points (1, 1) and (1, -1).
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 3.9
Find the derivatives of the functions in Exercises 1–42.
__
s = √ t .
1 + √ t
Verified step by step guidance1
Identify the function for which you need to find the derivative. In this case, the function is \( s = \sqrt{t} (1 + \sqrt{t}) \).
Rewrite the function in a more convenient form for differentiation. Express \( \sqrt{t} \) as \( t^{1/2} \), so the function becomes \( s = t^{1/2} (1 + t^{1/2}) \).
Apply the distributive property to expand the function: \( s = t^{1/2} + t^{1/2} \cdot t^{1/2} = t^{1/2} + t^{1} \).
Differentiate each term separately using the power rule. The power rule states that the derivative of \( t^n \) is \( n \cdot t^{n-1} \).
For the first term \( t^{1/2} \), the derivative is \( \frac{1}{2} t^{-1/2} \). For the second term \( t^1 \), the derivative is \( 1 \cdot t^{0} = 1 \). Combine these results to find the derivative of the function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are denoted as f'(x) or dy/dx.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function g(x) that is composed with another function f(u), the derivative is found by multiplying the derivative of the outer function by the derivative of the inner function. This is essential when differentiating functions that involve square roots or other composite forms.
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Power Rule
The power rule is a basic rule for finding the derivative of a function in the form of f(x) = x^n, where n is any real number. According to this rule, the derivative is f'(x) = n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions and functions involving roots, making it easier to find derivatives quickly.
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Related Practice
Textbook Question
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
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
s = cos⁴ (1 - 2t)
Textbook Question
Linearization for Approximation
In Exercises 7–12, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.
f(x) = ∛x, a = 8.5
Textbook Question
Find the derivatives of the functions in Exercises 19–40.
y = (1 / 18)(3x − 2)⁶ + (4 − (1 / 2x²))⁻¹
Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
r = 12/θ − 4/θ³ + 1/θ⁴