How do you find the derivative of #cos^-1(-1/x)# ?

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Answer 1 · 2018-05-15T15:58:36

#d/(dx)cos^(-1)(-1/x)=-1/(xsqrt(x^2-1)#

As #d/(dx)cos^(-1)x=-1/sqrt(1-x^2)# See detail here.

Now let #f(x)=cos^(-1)(-1/x)# and #g(x)=(-1/x)#

then #f(x)=cos^(-1)(g(x))# and using chain rule

#(df)/(dx)=-1/sqrt(1-g(x)^2)xx(dg)/(dx)#

= #-1/sqrt(1-(-1/x)^2)xxd/(dx)(-1/x)#

= #-1/sqrt(1-1/x^2)xx(-(-1/x^2))#

= #-x/sqrt(x^2-1)xx1/x^2#

= #-1/(xsqrt(x^2-1)#