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

Answer 1

Abigail Armstrong

#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)#

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Answer 2

Oliver Atkinson

To find the derivative of cos^(-1)(-1/x), you can use the chain rule. The derivative is -(1/sqrt(1 - (-1/x)^2)) * (d/dx)(-1/x), which simplifies to -(1/sqrt(1 - 1/x^2)) * (1/x^2).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.