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(a) One way of defining $ \sec^{-1} x $ is to say that $ y= \sec^{-1} x \Leftrightarrow \sec y = x $ and $ 0 \le y < \pi/2 $ or $ \pi \le y < 3\pi/2. $ Show that, with this definition,$ \frac {d}{dx} (\sec^{-1} x) = \frac {1}{x \sqrt {x^2 - 1}} $(b) Another way of defining $ \sec^{-1} x \Leftrightarrow \sec y = x $ and $ 0 \le y \le \pi, y \not= \pi/2. $ Show that, with this definition,$ \frac {d}{dx} (\sec{-1} x) = \frac {1}{\mid x \mid \sqrt {x^2 -1}} $

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Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Campbell University

Oregon State University

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.

04:15

(a) One way of defining $\…

01:31

The inequality sec $x \geq…

03:22

Okay, so the derivative of r c connects basically the set up is the same. You start by writing it up and you take implicit derivative and definitely, as the rule they write y prime in terms of x. However, there's a startle issue here depends on how you pick your domain and the and the range where you are defining the inverse function, because, in order to define the inverse function, you must have a 1 to 1 function and and if you want to find you Need to have 11 function depends on how you pick your domain and the and this tangent of arc connex might be off by a negative sign, as is indicated here and i'll. Let you figure that you gonna look at a lot of trignometry, either precocious text book or just to work out or get out how the sign change. So, when you pick a different domain to define or arrange to define the inverse function,

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