Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 8x; a = −3
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 3.10.7a
Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
a. (f^-1)'(4)
Verified step by step guidance1
Step 1: Understand the problem. We need to find the derivative of the inverse function \((f^{-1})'(4)\). This involves using the formula for the derivative of an inverse function.
Step 2: Recall the formula for the derivative of an inverse function. If \(y = f^{-1}(x)\), then \((f^{-1})'(x) = \frac{1}{f'(y)}\) where \(y = f^{-1}(x)\).
Step 3: Identify the value of \(y\) such that \(f(y) = 4\). This means we need to find \(y\) from the table where \(f(y) = 4\).
Step 4: Once \(y\) is identified, find \(f'(y)\) from the table. This is the derivative of \(f\) at the point \(y\).
Step 5: Use the formula \((f^{-1})'(4) = \frac{1}{f'(y)}\) to find the derivative of the inverse function at \(x = 4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
Inverse functions are functions that 'reverse' the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes y back to x. Understanding how to find and work with inverse functions is crucial for determining their derivatives.
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Inverse Cosine
Derivative of Inverse Functions
The derivative of an inverse function can be calculated using the formula (f^-1)'(y) = 1 / f'(x), where y = f(x). This relationship shows that the derivative of the inverse at a point is the reciprocal of the derivative of the original function at the corresponding point. This concept is essential for solving problems involving the derivatives of inverse functions.
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07:26Derivatives of Inverse Sine & Inverse Cosine
Using Tables for Derivatives
When working with derivatives from a table, it is important to locate the relevant values for the function and its inverse. The table typically provides values of the function and its derivative at specific points, which can be used to find the necessary derivatives of the inverse function. Understanding how to interpret and extract information from such tables is key to solving derivative problems.
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Related Practice
Textbook Question
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w
Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √2x+1; a= 4
Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4
Textbook Question
City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.
a. Compute A'(t). What units are associated with this derivative and what does the derivative measure?
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Textbook Question
Find an equation of the line tangent to the following curves at the given value of x.
y = csc x; x = π/4