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Find the numerical value of each expression.(a) $ \sinh 1 $ (b) $ \sinh^{-1} 1 $

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$$\sin h^{-1} 1 \approx 0.881$$

00:27

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

Derivatives

Differentiation

Campbell University

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

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.

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Find the numerical value o…

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Find the exact value of ea…

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to find the hyperbolic sine of one. We note that the hyperbolic sine of X. This is equal to erase the x minus erase the negative X over two. And so if Taxes one We have Hyperbolic Sine of one. This is equal to erase to one minus erased negative 1/2. That's the same as 1/2 times E minus one over E. Or this is equal to 1/2 times t squared minus one over E. Or that's the same as E squared minus one over to E. Now for the inverse of the hyperbolic sine suppose we let this as X. And so from here we have one equal to hyperbolic sine of X. Now hyperbolic side effects. This is suggests erased X minus erase the negative X over two. And so from here we have one equal to e. race to X erase the negative X over two. And then from here we will solve for X. Multiplying both sides by two. We have to equal to the race to x minus one over e. to the zero equal to erase x squared -2 E. raised X -1. From here. We want to apply quadratic formula and we have erase two X. This is equal to negative B plus or minus the square root of B squared minus four. A. C over two A. In which it is just one And then B is -2 & C is -1. So we have erase excess is equal to Negative of -2. That's positive too plus or minus. You have negative two square. That's for minus four times one times negative one Over two times 1. This is just too Plus or minus the square it of eight over to And it's just just equal to two plus or minus two squared of 2/2. Or that's just one plus or minus squared of two. Now, if this is our erased X. Then applying Ellen on both sides, we have L. N. Of one plus or minus the square root of two. This is equal to X. Therefore the hyperbolic sine or the inverse of the hyperbolic sine of one. This is equal to Ln of one plus the square root of two. Not that we cannot include the negative because 1- Creative to will be a negative value inside the natural log. So this is just the value of the inverse of the hyperbolic sine.

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