derivatives of inverse trigonometry functions

iiharthero · #1 $Old$ November 2nd, 2009, 03:51 AM

Find the derivatives of the following:

1) f(x) = cos-1(1/x-1)
2) f(x) = sin-1(2x+1)
3) f(x) = sin-1(pi/x)

thankyou for any help given !!

earboth · #2 $Old$ November 2nd, 2009, 04:11 AM

Quote:

Originally Posted by iiharthero $View Post$

Find the derivatives of the following:

1) f(x) = cos-1(1/x-1)
2) f(x) = sin-1(2x+1)
3) f(x) = sin-1(pi/x)

thankyou for any help given !!

In all 3 questions you have to use the chain rule. I'll show you how to do a) and leave the rest for you:

1. $f(x)=\arccos(x)~\implies~f'(x)=\dfrac{-1}{\sqrt{1-x^2}}$

$f(x)=\arccos\left(\dfrac1x-1\right)~\implies~f'(x)=\dfrac{-1}{\sqrt{1-\left(\dfrac1x-1\right)^2}} \cdot \left(-\dfrac1{x^2}\right)$

2. Simplify:

$\dfrac{-1}{\sqrt{1-\dfrac1{x^2}+\dfrac2x-1}} \cdot \left(-\dfrac1{x^2}\right) = \dfrac1{\dfrac1{|x|} \cdot \sqrt{2x-1}} \cdot \dfrac1{x^2} = \dfrac1{|x| \cdot \sqrt{2x-1}}$