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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.3.2
Capitolo 3, Problema 3.3.2

Derivative Calculations


In Exercises 1–12, find the first and second derivatives.


y = x² + x + 8

Guida verificata passo dopo passo
1
Step 1: Identify the function for which you need to find the derivatives. In this case, the function is y = x² + x + 8.
Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that if y = xⁿ, then the derivative dy/dx = n*x^(n-1).
Step 3: Apply the power rule to the first term x². The derivative of x² is 2*x^(2-1) = 2x.
Step 4: Apply the power rule to the second term x. The derivative of x is 1*x^(1-1) = 1.
Step 5: The constant term 8 has a derivative of 0, as the derivative of any constant is zero. Combine the derivatives of each term to find the first derivative: dy/dx = 2x + 1. Then, apply the same process to find the second derivative by differentiating dy/dx = 2x + 1 again.

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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. In practical terms, the derivative provides the slope of the tangent line to the curve of the function at any given point.
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Derivatives

Power Rule

The Power Rule is a fundamental technique for finding derivatives of polynomial functions. It states that if a function is in the form of f(x) = x^n, where n is a real number, then its derivative f'(x) is given by f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for functions involving powers of x.
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Power Rules

Second Derivative

The second derivative of a function is the derivative of the first derivative. It provides information about the curvature of the function's graph and can indicate whether the function is concave up or concave down. The second derivative is useful for analyzing the acceleration of a function and identifying points of inflection where the function changes concavity.
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The Second Derivative Test: Finding Local Extrema
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Domanda del libro di testo

[Technology Exercise]


Graph the curves in Exercises 39–48.

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.


y = 4x²/⁵ − 2x

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Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?

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Domanda del libro di testo

In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?


54. x³ + y² = sin²y

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Finding Linearizations


In Exercises 1–5, find the linearization L(x) of f(x) at x = a.


f(x) = ∛x, a = −8

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Domanda del libro di testo

Derivatives in Differential Form


In Exercises 17–28, find dy.


xy² − 4x³/² − y = 0

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Domanda del libro di testo

Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².

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