Textbook Question
Use the graph of g in the figure to do the following. <IMAGE>
b. Find the values of x in (-2,2) at which g is not differentiable.
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 3.10.61b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. d/dx(tan^−1 x) =sec² x
Verified step by step guidance1
To determine whether the statement \( \frac{d}{dx}(\tan^{-1} x) = \sec^2 x \) is true, we need to find the derivative of \( \tan^{-1} x \).
Recall that \( \tan^{-1} x \) is the inverse function of \( \tan x \). The derivative of \( \tan^{-1} x \) is given by the formula \( \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2} \).
Compare the derivative \( \frac{1}{1 + x^2} \) with \( \sec^2 x \). Note that \( \sec^2 x = 1 + \tan^2 x \), which is different from \( \frac{1}{1 + x^2} \).
Since \( \frac{1}{1 + x^2} \) is not equal to \( \sec^2 x \), the statement \( \frac{d}{dx}(\tan^{-1} x) = \sec^2 x \) is false.
Therefore, the correct derivative of \( \tan^{-1} x \) is \( \frac{1}{1 + x^2} \), not \( \sec^2 x \). This serves as a counterexample to the given statement.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Inverse Functions
The derivative of an inverse function can be found using the formula (d/dx)(f^−1(x)) = 1/(f'(f^−1(x))). For the function f(x) = tan(x), its inverse is f^−1(x) = tan^−1(x). Understanding this relationship is crucial for differentiating inverse trigonometric functions like tan^−1(x).
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07:26
Derivatives of Inverse Sine & Inverse Cosine
Trigonometric Derivatives
Knowing the derivatives of basic trigonometric functions is essential. For example, the derivative of tan(x) is sec²(x). This knowledge helps in finding the derivative of its inverse, tan^−1(x), and is fundamental in verifying the correctness of derivative statements.
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06:35Derivatives of Other Inverse Trigonometric Functions
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). This rule is particularly useful when dealing with functions like tan^−1(x) that can be expressed in terms of other functions, aiding in the differentiation process.
Recommended video:
05:02Intro to the Chain Rule
Related Practice
Textbook Question
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.
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Textbook Question
45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
x³+y³=2xy; (1, 1)
Textbook Question
The energy (in joules) released by an earthquake of magnitude M is given by the equation E=25,000 ⋅ 101.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)
Compute dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)
Textbook Question
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
4x³ =y²(4−x); x=2 (cissoid of Diocles)
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Textbook Question
{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.
b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.