`y = tanh^-1(x)+ ln(sqrt(1-x^2))` Find the derivative of the function

Given,

$\displaystyle{y}={{\tanh}}^{{-{{1}}}}{\left({x}\right)}+{\ln{{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}}}$

so we have to find the y'

so,

$\displaystyle{y}'={\left({{\tanh}}^{{-{{1}}}}{\left({x}\right)}+{\ln{{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}}}\right)}'$

$\displaystyle={\left({{\tanh}}^{{-{{1}}}}{\left({x}\right)}\right)}'+{\left({\ln{{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}}}\right)}'$

as we know

$\displaystyle{\left({{\tanh}}^{{-{{1}}}}{\left({x}\right)}\right)}'=\frac{{1}}{{{1}-{{x}}^{{2}}}}$

and so we have to find out

$\displaystyle{\left({\ln{{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}}}\right)}'$

let $\displaystyle{u}=\sqrt{{{1}-{{x}}^{{2}}}}$

it can be solved by the following,

$\displaystyle\frac{{{d}{f}}}{{\left.{d}{x}}}=\frac{{{d}{f}}}{{{d}{u}}}\cdot\frac{{{d}{u}}}{{\left.{d}{x}}}$

so,

$\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left({\ln{{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}}}\right)}=\frac{{d}}{{{d}{u}}}{\ln{{\left({u}\right)}}}\cdot\frac{{{d}{u}}}{{\left.{d}{x}}}$

$\displaystyle=\frac{{1}}{{u}}\cdot\frac{{d}}{{\left.{d}{x}}}{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}$

$\displaystyle=\frac{{1}}{\sqrt{{{1}-{{x}}^{{2}}}}}\cdot{\left(-\frac{{x}}{\sqrt{{{1}-{{x}}^{{2}}}}}\right)}$

$\displaystyle=-\frac{{x}}{{{1}-{{x}}^{{2}}}}$

now,as

$\displaystyle{y}'={\left({{\tanh}}^{{-{{1}}}}{\left({x}\right)}+{\ln{{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}}}\right)}'$

$\displaystyle=\frac{{1}}{{{1}-{{x}}^{{2}}}}+{\left(-\frac{{x}}{{{1}-{{x}}^{{2}}}}\right)}$

$\displaystyle=\frac{{{1}-{x}}}{{{1}-{{x}}^{{2}}}}$

$\displaystyle=\frac{{1}}{{{1}+{x}}}$

Blog

Monday, June 8, 2009

$\displaystyle{y}={{\tanh}}^{{-{{1}}}}{\left({x}\right)}+{\ln{{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}}}$ Find the derivative of the function

No comments:

Post a Comment

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Monday, June 8, 2009

Find the derivative of the function

No comments:

Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{y}={{\tanh}}^{{-{{1}}}}{\left({x}\right)}+{\ln{{\left(\sqrt{{{1}-{{x}}^{{2}}}}\right)}}}$ Find the derivative of the function

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?