SOLUTION: Sorry I can’t find calculus topic so I’ll ask here instead, am I able to find the derivative of y=arcsec x by using {{{ 1/x = cos y }}}? But it turns out that the derivative i

Algebra ->  Trigonometry-basics -> SOLUTION: Sorry I can’t find calculus topic so I’ll ask here instead, am I able to find the derivative of y=arcsec x by using {{{ 1/x = cos y }}}? But it turns out that the derivative i      Log On


   

Question 1178008: Sorry I can’t find calculus topic so I’ll ask here instead, am I able to find the derivative of y=arcsec x
by using +1%2Fx+=+cos+y+? But it turns out that the derivative is wrong and I obtain +dy%2Fdx=+1%2F%28x%5E2+%2Asqrt%28x%5E2-1%29%29+
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
find the derivative of y=arcsec%28+x%29+by using 1%2Fx=cos%28y%29

y=arcsec%28x%29=1%2Farccos+%28x%29=arccos+%281%2Fx%29
let 1%2Fx=u

%28d%2Fdx%29%28arccos%28u%29%29=1%2F%28sqrt%281-%281%2Fx%29%29%5E2%29*u’

%28dy%2Fdx%29=%28-1%2F%28sqrt%281-%281%2Fx%29%29%5E2%29%29%2A%28-1%2Fx%5E2%29
%28dy%2Fdx%29=1%2F%28x%5E2%2Asqrt%281-%281%2Fx%29%29%5E2%29

you obtain +dy%2Fdx=+1%2F%28x%5E2+%2Asqrt%28x%5E2-1%29%29+, and this part sqrt%28x%5E2-1%29 is incorrect , suppose to be sqrt%281-x%5E2%29



Answer by ikleyn(52884) About Me  (Show Source):
You can put this solution on YOUR website!
.

From  y = arcsec(x), you have


    sec(y) = x


and then, step by step


    1%2Fcos%28y%29 = x

    cos(y) = 1%2Fx

    d ( cos(y) ) = d ( 1%2Fx )

    -sin(y)*dy = - dx%2Fx%5E2

    sin(y)*dy = dx%2Fx%5E2

    %28dy%29%2F%28dx%29 = 1%2F%28x%5E2%2Asin%28y%29%29%29

    %28dy%29%2F%28dx%29%29 = 1%2F%28x%5E2%2Asqrt%281-cos%5E2%28y%29%29%29 = 1%2F%28x%5E2%2Asqrt%281-%281%2Fx%5E2%29%29%29 = x%2F%28x%5E2%2Asqrt%28x%5E2-1%29%29 = 1%2F%28x%2Asqrt%28x%5E2-1%29%29.


ANSWER.  %28dy%29%2F%28dx%29 = 1%2F%28x%2Asqrt%28x%5E2-1%29%29.

Solved.