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Show that $ \frac {d}{dx} \sqrt[4]{ \frac {1 + \tanh x}{1 - \tanh x}} = \frac {1}{2} e^{x/2}. $

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Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

Derivatives

Differentiation

Campbell University

Oregon State 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.

03:30

prove d/dx [ tanh(x)] …

02:29

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02:54

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Verify the integration for…

03:01

Use the fact that (tanh x)…

we know that sign h of X over coastline Ajax either the ex minister to the night of axe over each of the AKs, possibly to the negative axe gives us the four through of one plus tangent. A jukebox over one minus tangent. H rocks gives us the four through of to eat of the axe over to eat of the negative axe. Obviously, we can cancel out the twos. The 4th 3rd of each of the two acts equals e the 1/2 acts using the chain rule Eat of the X over two times 1/2. Remember, the derivative of X over two is 1/2 gives us 1/2 eat of the X over too.

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