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Prove the identity. $ \sinh (-x) = -\sinh x $ (This shows that $ \sinh $ is an odd function.)

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00:20

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

Derivatives

Differentiation

Campbell University

University of Nottingham

Idaho 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.

01:23

The hyperbolic sine functi…

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$7-19$ Prove the identity.…

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Prove that $\sinh x$ is an…

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Verify that the hyperbolic…

01:04

Verify the identity.$$\sin…

02:36

Show that $\sinh (-x)=-\si…

05:31

The hyperbolic cosine and …

All right, So in this problem, we are going to prove sinche. The hyperbolic Sinus sometimes pronounce inch. So for simplicity, I'm going to call it that way. We're going to prove cinch off. Negative X is negative. Sin jacks. Now, if you can recall that f off negative, X equaling effort negative ffx when this happens, F of X is called an odd function. So this problem is equivalent to saying, since of X is an odd function, and that's what we're going to prove. All right, so we start the proof with the definition. What? It's sinche FX. We know that this is equal to e to the X minus e to the negative x divided by two. So to figure out what cinch off negative X is equal to, we substitute X with negative X and see what's gonna happen. The first X becomes a negative x negative X becomes a positive X, And as you can see, you can pull out a negative from both of these two terms on the numerator. So it'll be negative e to the x minus into the matrix, all divided by two. And we know that this portion is cinch of X, so it to sum it up, this is negative cinch of X, and that is exactly what we're trying to prove. So to summarize it, we start with the definition we plug in Negative X and then showed that we are able to pull out a negative, and that proves that cinch of X is an odd function.

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