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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.6.75f
Capitolo 3, Problema 3.6.75f

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.


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Find the derivatives with respect to x of the following combinations at the given value of x.


f. √f(x), x = 2

Guida verificata passo dopo passo
1
Step 1: Recall the chain rule for derivatives. If h(x) = √f(x), then h'(x) = (1 / (2√f(x))) * f'(x). This formula is derived by differentiating the square root function and applying the chain rule.
Step 2: Identify the values of f(x) and f'(x) at x = 2 from the table. From the table, f(2) = 8 and f'(2) = 1/3.
Step 3: Substitute f(2) and f'(2) into the derivative formula h'(x) = (1 / (2√f(x))) * f'(x). This gives h'(2) = (1 / (2√8)) * (1/3).
Step 4: Simplify the expression. The square root of 8 can be written as √8 = 2√2. Substitute this into the formula to simplify further.
Step 5: Combine the constants and simplify the fraction to express the derivative in its simplest form. Do not calculate the final numerical value, as the focus is on the process.

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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 this context, knowing the derivatives of functions f and g at specific points is crucial for finding the derivatives of their combinations.
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Derivatives

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y is composed of two functions u and v, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential for differentiating functions like √f(x), where f(x) is itself a function.
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Intro to the Chain Rule

Square Root Function

The square root function, denoted as √f(x), is a function that returns the non-negative square root of f(x). When differentiating this function, it is important to apply the chain rule, as the derivative of √u (where u = f(x)) involves the derivative of f(x) as well. Understanding how to differentiate square root functions is key to solving the given problem.
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Multiplying & Dividing Functions
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Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


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Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.


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Find the derivatives with respect to x of the following combinations at the given value of x.


f. (x¹¹ + f(x))⁻², x = 1

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Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


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s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

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Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.


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Find the derivatives with respect to x of the following combinations at the given value of x.


g. f(x + g(x)), x = 0

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Lunar projectile motion A rock thrown vertically upward from the surface of the moon at a velocity of 24 m/sec (about 86 km/h) reaches a height of s = 24t − 0.8t² m in t sec.

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Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)

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Find the first derivatives of the following combinations at the given value of x.


g. ƒ(x + g(x)), x = 0

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