Skip to main content
Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.6.75f
Chapter 3, Problem 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.


<IMAGE>


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


f. √f(x), x = 2

Verified step by step guidance
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

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. In this context, knowing the derivatives of functions f and g at specific points is crucial for finding the derivatives of their combinations.
Recommended video:
05:44
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.
Recommended video:
05:02
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.
Recommended video:
7:24
Multiplying & Dividing Functions
Related Practice
Textbook Question

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:


e. When is it moving fastest (highest speed)? Slowest?


s = 4 - 7t + 6t² - t³, 0 ≤ t ≤ 4

Textbook Question

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


" style="max-width: 100%; white-space-collapse: preserve;" width="250">


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


f. (x¹¹ + f(x))⁻², x = 1

Textbook Question

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:


f. When is it farthest from the axis origin?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

Textbook Question

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:


d. When does it speed up and slow down?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

Textbook Question

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.

e. How long is the rock aloft?

Textbook Question

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)

0 1 1 -3 1/2

1 3 5 1/2 -4


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


g. ƒ(x + g(x)), x = 0