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 91b
If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
b. (f^-1)'(3)
Verified step by step guidance1
Understand the problem: We need to find the derivative of the inverse function at a specific point, (f^-1)'(3). This involves using the relationship between a function and its inverse.
Recall the formula for the derivative of an inverse function: If y = f(x) and f is invertible, then the derivative of the inverse function at a point is given by (f^-1)'(y) = 1 / f'(x), where x = f^-1(y).
Identify the point on the graph: We need to find the x-value such that f(x) = 3. This means we are looking for the x-coordinate where the function f has a value of 3.
Use the graph of f to find the x-value: Locate the point on the graph of f where the y-coordinate is 3. This x-value will be used in the formula for the derivative of the inverse function.
Evaluate the derivative using the graph of f': Once the x-value is found, use the graph of f' to determine f'(x). Then, apply the formula (f^-1)'(3) = 1 / f'(x) to find the derivative of the inverse function at the 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. This theorem provides a way to find the derivative of the inverse function using the formula (f^-1)'(y) = 1 / f'(f^-1(y)), which is essential for evaluating derivatives of inverse functions.
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Inverse Cosine
Derivative of a Function
The derivative of a function measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Understanding how to compute and interpret derivatives is crucial for analyzing the behavior of functions and their inverses.
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06:30Derivatives of Other Trig Functions
Graphical Interpretation of Derivatives
The graphical interpretation of derivatives involves understanding how the slope of the tangent line to the graph of a function at a given point represents the derivative at that point. For inverse functions, the slopes of the original function and its inverse are reciprocals at corresponding points, which is a key concept when evaluating derivatives using graphs.
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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
If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
a. (f^-1)'(7)
1
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Textbook Question
Derivatives from tangent lines 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.
b. y = f(x) / g(x)
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.