Textbook Question
Tangent lines Determine an equation of the line tangent to the graph of y=(x²−1)² / x³−6x−1 at the point (0,−1).
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 91a
If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
a. (f^-1)'(7)
Verified step by step guidance1
To find the derivative of the inverse function at a point, we use the formula: \((f^{-1})'(b) = \frac{1}{f'(a)}\), where \(f(a) = b\).
Identify the point \(b = 7\) on the graph of \(f\). Find the corresponding \(a\) such that \(f(a) = 7\).
Once you have found \(a\), locate \(f'(a)\) on the graph of \(f'\). This is the slope of the tangent to \(f\) at \(a\).
Substitute \(f'(a)\) into the formula \((f^{-1})'(7) = \frac{1}{f'(a)}\) to find the derivative of the inverse function at 7.
Ensure that \(f'(a) \neq 0\) to avoid division by zero, which would indicate that the inverse function is not differentiable at that point.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Function Theorem
The Inverse Function Theorem states that if a function f is continuous and differentiable, and its derivative f' is non-zero at a point, then the inverse function f^-1 exists locally around that point. The derivative of the inverse function can be calculated using the formula (f^-1)'(y) = 1 / f'(f^-1(y)), which relates the derivatives of the function and its inverse.
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Derivative Interpretation
The derivative of a function at a point represents the slope of the tangent line to the graph of the function at that point. In the context of the question, understanding how to interpret the derivative graphically is crucial for evaluating (f^-1)'(7), as it involves analyzing the behavior of f and its inverse at specific values.
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Graphical Analysis of Functions
Graphical analysis involves examining the graphs of functions and their derivatives to understand their behavior. For the given question, one must analyze the graph of f to find the corresponding x-value for f(x) = 7, and then use the graph of f' to determine the slope at that point, which is essential for calculating the derivative of the inverse function.
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Related Practice
Textbook Question
Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.
y = f(x)g(x)
Textbook Question
Second derivatives Find d²y/dx²for the following functions.
y = e^-2x²
Textbook Question
If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
b. (f^-1)'(3)
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Textbook Question
{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the tangent lines.
Textbook Question
Derivatives by different methods
a. Calculate d/dx (x²+x)² using the Chain Rule. Simplify your answer.