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Prove the formulas given in Table 6 for the derivatives of the following functions. (a) $ \cosh^{-1} $ (b) $ \tanh^{-1} $ (c) $ csch^{-1} $(d) $ sech^{-1} $ (e) $ \coth^{-1} $

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Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

Derivatives

Differentiation

Baylor University

University of Michigan - Ann Arbor

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.

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Prove the formulas given i…

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Prove each of the differen…

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Find the derivative of eac…

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Find the derivatives of $\…

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he It's clear it's only me right here So that inverse of co sign for my public function. It is equal to L on of X plus square root of X square minus one, and for the derivative L N of X plus the square root of X square minus one, you have it equal to one over X plus square. Root that square minus one tons de over D x of X plus square root of X square minus one, which is equal to one over X plus square root specs Square minus one win won't supply square root of X square minus one waas X over square of X square minus one, which is equal to one over square root of X square minus one part B where you make why we go up to the inverse tangent. The hyperbolic function so tangent of why is equal to X. We differentiate to get seeking square of nuts. So why d Why is equal to D. X? So we get D y o ver de acts to be won over the sequence square. So do you. Why, over DX is equal to one over one minus and Tangent Square which is equal to one over one minus expert for part c. Well, then allow Why to be able to cool seeker in verse when we get coast seeking is equal to X. We differentiate implicitly some negative co sika. Why co tangent public function of why do you Why is equal to d X? So we get you on a value for d y o ver de acts. We just stopped it too, which becomes equal to negative one all over co sika wide plus or minus squaring of one plus co Siegen square of one which is equal to a negative one over X times plus or minus square root of one plus X square which is equal to negative one over absolute value of X square root of one plus X squared. For a party, we allow why to be going to seek int in verse. So seeking of why is equal to X. So we got a deal over D X to be equal to negative one over a secret over tangent. We're just gonna substitute and just the hyperbaric function. Why? For plus or minus square root of one minus see good square. So we got negative one over my public function of sequined, plus or minus square root one minus sequence square of why and this becomes equal to negative one over X plus or minus. It's where it of one minus X square. So looking at the domain, we could just use the positive route since it zero comma negative zero comma, one included.

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