Textbook Question
Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?
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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 3.6.68
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1
Verified step by step guidance1
First, understand that you need to find the derivative of the composition of two functions, (f ∘ g)(x), which is f(g(x)).
Identify the functions involved: f(u) = 1 - (1/u) and g(x) = 1 / (1 - x).
Apply the chain rule for derivatives, which states that the derivative of a composition of functions (f ∘ g)'(x) is f'(g(x)) * g'(x).
Calculate g'(x) by differentiating g(x) = 1 / (1 - x). Use the quotient rule or recognize it as a power function to find g'(x).
Calculate f'(u) by differentiating f(u) = 1 - (1/u). Use the power rule to find f'(u). Then evaluate f'(g(x)) at x = -1 using the values obtained from the previous steps.
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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. It states that if you have a function f(g(x)), the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is essential for finding the derivative of functions where one function is nested within another.
Recommended video:
05:02
Intro to the Chain Rule
Composite Functions
A composite function is formed when one function is applied to the result of another function, denoted as f(g(x)). Understanding how to work with composite functions is crucial for applying the Chain Rule effectively. In this problem, f(u) and g(x) are composite, where g(x) serves as the input for f.
Recommended video:
3:48Evaluate Composite Functions - Special Cases
Evaluating Derivatives
Evaluating derivatives involves calculating the slope of a function at a specific point. In this context, after applying the Chain Rule, you will need to substitute the given value of x into the derivative expression to find the specific value of (f ∘ g)' at that point. This step is critical for obtaining the final answer.
Recommended video:
5:14Evaluate Logarithms
Related Practice
Textbook Question
Derivatives
In Exercises 1–18, find dy/dx.
f(x) = x³ sin x cos x
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Textbook Question
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
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Textbook Question
Derivatives
In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
244
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Textbook Question
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
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Textbook Question
In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
y = 4 − x², (−1, 3)
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