`f(t) = arctan(sinht)` Find the derivative of the function

Monday, September 6, 2010

$\displaystyle{f{{\left({t}\right)}}}={\arctan{{\left({\sinh{{t}}}\right)}}}$ Find the derivative of the function

$\displaystyle{f{{\left({t}\right)}}}={\arctan{{\left({\sinh{{\left({t}\right)}}}\right)}}}$

Take note that the derivative formula of arctangent is

$\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[{\arctan{{\left({u}\right)}}}\right]}=\frac{{1}}{{{1}+{{u}}^{{2}}}}\cdot\frac{{{d}{u}}}{{\left.{d}{x}}}$

Applying this, the derivative of the function will be

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{d}}{{{\left.{d}{t}\right.}}}{\left[{\arctan{{\left({\sinh{{\left({t}\right)}}}\right)}}}\right]}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{1}}{{{1}+{{\sinh}}^{{2}}{\left({t}\right)}}}\cdot\frac{{d}}{{{\left.{d}{t}\right.}}}{\left[{\sinh{{\left({t}\right)}}}\right]}$

Also, the derivative formula of hyperbolic sine is

$\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[{\sinh{{\left({u}\right)}}}\right]}={\cosh{{\left({u}\right)}}}\cdot\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}$

Applying this, f'(t) will become

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{1}}{{{1}+{{\sinh}}^{{2}}{\left({t}\right)}}}\cdot{\cosh{{\left({t}\right)}}}\cdot\frac{{d}}{{{\left.{d}{t}\right.}}}{\left({t}\right)}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{1}}{{{1}+{{\sinh}}^{{2}}{\left({t}\right)}}}\cdot{\cosh{{\left({t}\right)}}}\cdot{1}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{\cosh{{\left({t}\right)}}}}{{{1}+{{\sinh}}^{{2}}{\left({t}\right)}}}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{\cosh{{\left({t}\right)}}}}{{{{\cosh}}^{{2}}{\left({t}\right)}}}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{1}}{{\cosh{{\left({t}\right)}}}}$ is the final derivative

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Monday, September 6, 2010

$\displaystyle{f{{\left({t}\right)}}}={\arctan{{\left({\sinh{{t}}}\right)}}}$ Find the derivative of the function

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

Monday, September 6, 2010

Find the derivative of the function

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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?

$\displaystyle{f{{\left({t}\right)}}}={\arctan{{\left({\sinh{{t}}}\right)}}}$ 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?