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Prove the identity.$ \tanh (\ln x) = \frac {x^2 - 1}{x^2 + 1} $

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Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

Baylor University

Boston College

Lectures

04:40

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.

44:57

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.

01:46

Prove that $\tanh ^{-1}\le…

02:44

03:01

Prove that Tanh(ln(x)) …

03:21

Prove the identity.$$<…

01:57

Prove the identity.$$\…

01:10

Prove the identity.$ \…

02:35

Prove the identity.

03:10

02:50

Verify the identity.$$…

03:00

Verify the identity.$\…

02:40

Prove that $\tanh ^{-1} x=…

01:34

Verify each identity.$…

We know that tangent is sign over cosine. This is equivalent to e to the natural log of x. Minus the negative natural log of x over e to the natural log of x plus e, is the negative natural log of x is equivalent to x, squared over x squared. We have x, squared minus 1 over x, squared plus 1 point. Therefore, we know that sine h of x is e of x and c of negative x over 2 cosine h of x e of x, plus e of negative x over 2, and then tangent we know is sine divided by cosine.

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