Derivative of arcsech
( \newcommand{\kernel}{\mathrm{null}\,}\)
Derivative of sech-1(x)
We use the fact from the definition of the inverse that
sech(sech−1x)=x
and the fact that
sech′x=−tanh(x)sech(x)
Now take the derivative of both sides (using the chain rule on the left hand side) to get
−tanh(sech−1x)sech(sech−1x)(sech−1x)′=1
or
−xtanh(sech−1x)(sech−1x)′=1
We know that
cosh2x−sinh2x=1
Dividing by the cosh2(x) gives
1−tanh2(x)=sech2x
or
tanhx=√1−sech2x
so that
tanh(sech−1x)=√1−sech−1x=√1−x2
Finally substituting into equation 1 gives
−x√1−x2(sech−1x)=1
sech−1x=−1x√1−x2
Larry Green (Lake Tahoe Community College)