Textbook Question
Derivatives using tables Let and . Use the table to compute the following derivatives.
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b.
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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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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.
Recommended video:
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
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
³√x+³√y⁴ = 2;(1,1)
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Textbook Question
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
(x+y)^2/3=y; (4, 4)
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Textbook Question
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = √3x; a= 12
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Textbook Question
A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.
b. Show that y = B cos t satisfies the equation for any constant B.
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Textbook Question
{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.
b. Graph the volume function and verify that the tank is empty when t=1000 min.
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