What is the derivative of arcsec(x2)?

1 Answer
Jul 29, 2018

ddx(arcsec(x2))=2|x|x24

Explanation:

Let y=arcsec(x2), then:

2secy=x

Differentiate implicitly:

2secytanydydx=1

dydx=12secy1tany

dydx=1x1tany

using now the identity:

tan2y=sec2y1=x241=x244

we have that for x[2,+), that is for y[0,π2):

tany=x242, so:

dydx=2xx24

while for x(,2], that is for y(π2,π]:

tany=x242, so:

dydx=2xx24

We can then write the derivative for both intervals as.

dydx=2|x|x24