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Prove the formulas given in Table 6 for the deriv…

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Problem 28 Easy Difficulty

For each of the following function (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3.
(a) $ csch^{-1} $
(b) $ sech^{-1} $
(c) $ \coth^{-1} $


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Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

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Derivatives

Differentiation

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Video Thumbnail

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Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Problem 7
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Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
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Problem 18
Problem 19
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Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59

Video Transcript

okay. This problem, we're gonna use thes three formulas to get formulas for these three inverse, trig, hyperbolic trig functions. So we're going to start by setting Why, equal to the hyperbolic coast seeking inverse of X. Now take the hyperbolic coast seeking of both sides. So you get the hyperbolic coast. Second of y equals X. So, you know, the hyperbolic coast seeking is the same thing as one over the hyperbolic sign of why flip both these fractions upside down. So you get the hyperbolic sign of why is one over X Now take the hyperbolic sine inverse of both sides. All right, now we know what to do because we know a formula for the hyperbolic sine inverse. Everywhere you see X in the formula, put one over X. So this is equal to the natural log of one over X plus the square root of one over X squared plus one. All right, now we're going to find the formula for the hyperbolic seeking in verse what? We don't have to go through all those first steps because we can see that all you have to dio is put the reciprocal function and then the reciprocal um, argument in there. So you get the hyperbolic cosine inverse of one over X. Here's the formula. Fritz. Everywhere is he X Put one over X. So get the natural log of one over X plus the square root of one over X squared minus one. Okay. And then the hyperbolic co tangent in verse Why was the hyperbolic co tangent inverse of X is the reciprocal function so hyperbolic tangent in verse and the reciprocal argument one over X. So you get one half the natural log of 11 uh, of one plus one over X over one, minus one over X. Okay, you don't want to leave it like that with a four story fraction. So multiply everything in the fraction by X. It's like if you won half the natural log of X plus one over X minus one and there you formulas

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Calculus: Early Transcendentals

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Related Topics

Derivatives

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Top Calculus 1 / AB Educators
Catherine Ross

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Kayleah Tsai

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Michael Jacobsen

Idaho State University

Joseph Lentino

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Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
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