How do you find the derivative of the function: arcsec(x/2)?

1 Answer
sente
May 2, 2016

d/dx"arcsec"(x/2)=2/(xsqrt(x^2-4))

Explanation:

Using implicit differentiation, we start by letting y = "arcsec"(x/2)

=> sec(y) = x/2

=> d/dxsec(y) = d/dxx/2

=>sec(y)tan(y)dy/dx = 1/2

=> dy/dx = 1/(2sec(y)tan(y))

We already know sec(y) = x/2, and if we construct a right triangle with an angle y such that sec(y) = x/2 we find that tan(y) = sqrt(x^2-4)/2. Substituting these in, we have

d/dx"arcsec"(x/2) = dy/dx

= 1/(2(x/2)(sqrt(x^2-4)/2))

=2/(xsqrt(x^2-4))

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