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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.6.76g
Chapter 3, Problem 3.6.76g

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

Verified step by step guidance
1
Step 1: Recognize that the problem asks for the derivative of the composite function f(x + g(x)) at x = 0. To solve this, we will use the chain rule for derivatives.
Step 2: Recall the chain rule: If h(x) = f(u(x)), then h'(x) = f'(u(x)) * u'(x). Here, u(x) = x + g(x), so we need to compute the derivative of u(x) first.
Step 3: Compute u'(x). Since u(x) = x + g(x), its derivative is u'(x) = 1 + g'(x). At x = 0, g'(0) = 1/3, so u'(0) = 1 + 1/3 = 4/3.
Step 4: Apply the chain rule. The derivative of f(x + g(x)) at x = 0 is f'(u(0)) * u'(0). From the table, f'(x) at x = 0 is 5, and u(0) = 0 + g(0) = 0 + 1 = 1.
Step 5: Substitute the values into the chain rule expression. f'(u(0)) = f'(1), and from the table, f'(1) = -1/3. Combine this with u'(0) = 4/3 to complete the derivative calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. If you have a function h(x) = f(g(x)), the derivative h'(x) is found by multiplying the derivative of the outer function f at g(x) by the derivative of the inner function g at x, expressed as h'(x) = f'(g(x)) * g'(x). This rule is essential for solving problems involving nested functions.
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Intro to the Chain Rule

Function Composition

Function composition involves creating a new function by applying one function to the results of another. In the context of calculus, understanding how to compose functions is crucial for applying the chain rule. For example, if f(x) and g(x) are functions, the composition f(g(x)) means applying g first and then f to the result, which is key in differentiating composite functions.
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Evaluate Composite Functions - Special Cases

Derivative Evaluation

Derivative evaluation involves calculating the derivative of a function at a specific point. This requires substituting the given x-value into the derivative function. In the problem, you need to evaluate the derivative of f(x + g(x)) at x = 0, using the values provided in the table for f, g, and their derivatives at x = 0, ensuring accurate computation of the derivative at that point.
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Evaluate Logarithms
Related Practice
Textbook Question

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.


g. 1 / g²(x), x = 3

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.


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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

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

Finding Derivative Functions and Values 


Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.


p(θ) = √3θ; p′(1), p′(3), p′(2/3)

Textbook Question

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.


h. √(f²(x) + g²(x)), x = 2

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