Why absolute value is necessary in derivative of inverse secant

In this video, you will learn why the derivative of inverse secant has an absolute value? Why absolute value in derivative of inverse secant? You will also learn “how to find the derivative of secant inverse of x”. Non negative derivative of secant inverse x. Why is differentiation of sec inverse x always positive? Why is derivative of sec inverse x always positive? Why is derivative of inverse secant always positive? Why is the derivative of arcsec x always positive? Why is differentiation of sec inverse x contains mod x? Why is derivative of sec inverse x has mod x? Why is derivative of inverse secant has mod x? Why is the derivative of arcsec x has mod x? Why is differentiation of sec inverse x contains absolute value of x? Why is derivative of sec inverse x has absolute value of x? Why is derivative of inverse secant has absolute value of x? Why is the derivative of arcsec x has absolute value of x? Derivative of inverse secant x equals 1 over product of x and square root of x squared minus 1. The denominator of right-hand side contains two terms one is x and another is square root of x squared minus 1. The sign of the derivative will depend upon sign of x, because the quantity under square root is never negative. It means when x is positive the derivative or slope is positive and on other hand when x is negative the derivative or slope is negative.
But wait, this is not 100% true, this is just half story. Actually, the derivative of secant inverse of x contains absolute value of x not just x. When we analyse graph of secant inverse of x, the function is not defined for values of x between negative one and positive one. The slope of the function is not defined at 1 and negative 1. Also, when x is greater than or equal to 1; slope of this function is positive on the other hand when x is less than or equal to negative 1, then also slope is positive. So, to make the derivative always positive; we put absolute value sign. This is the reason why the derivative of inverse secant has an absolute value.
0:00 - Topics of lecture.
0:15 - Finding derivative of Inverse secant.
1:16 - Sign Analysis of derivative of inverse secant.
1:39 - Why the derivative of inverse secant has an absolute value?

To differentiate sec inverse of x; let y equals the secant inverse of x. This implies, secant of y equals x. Let us differentiate both sides with respect to y. The derivative of secant of y equals secant of y times tangent of y. Right hand side equals dx over dy. This implies, dy over dx equals 1 over product of secant of y and tangent of y. Now we will substitute tangent of y by square root of secant squared y minus 1. We have almost done it, Since, secant of y equals x, hence replace secant of y by x.