Why is the Absolute Value of X Taken in the Derivative of Arc Secant?

[wazzup]

Hello there

In the derivative of the arc secant, why is the absolute value of x ( which is present in the denominator) taken? Is this to prevent the possible of having a zero ( and making the whole expression undefined ? )


Thanks

 

[HallsofIvy]

Hello there

In the derivative of the arc secant, why is the absolute value of x ( which is present in the denominator) taken? Is this to prevent the possible of having a zero ( and making the whole expression undefined ? )


Thanks

No, absolute value won't do that: |0|= 0 so you still can have in the denominator. (And the derivative is undefined for x= 0.) Strictly speaking the inverse secant is only defined for x less than -1 or greater than 1. It is because of the square root in the derivative that we have a "plus or minus" which, combined with x itself, makes the derivative always positive.