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Derivative of arcsech

( \newcommand{\kernel}{\mathrm{null}\,}\)

Derivative of sech-1(x)

We use the fact from the definition of the inverse that

sech(sech1x)=x

and the fact that

sechx=tanh(x)sech(x)

Now take the derivative of both sides (using the chain rule on the left hand side) to get

tanh(sech1x)sech(sech1x)(sech1x)=1

or

xtanh(sech1x)(sech1x)=1

We know that

cosh2xsinh2x=1

Dividing by the cosh2(x) gives

1tanh2(x)=sech2x

or

tanhx=1sech2x

so that

tanh(sech1x)=1sech1x=1x2

Finally substituting into equation 1 gives

x1x2(sech1x)=1

sech1x=1x1x2

Larry Green (Lake Tahoe Community College)


This page titled Derivative of arcsech is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

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