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

Saturday, September 12, 2009

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

This is a composite function, and to differentiate it we need the chain rule,

$\displaystyle{\left({f{{\left({g{{\left({x}\right)}}}\right)}}}\right)}'={f{'}}{\left({g{{\left({x}\right)}}}\right)}\cdot{g{'}}{\left({x}\right)}.$

Here $\displaystyle{g{{\left({x}\right)}}}={\sin{{\left({2}{x}\right)}}}$ and $\displaystyle{f{{\left({z}\right)}}}={{\tanh}}^{{-{{1}}}}{\left({z}\right)},$ so we need their derivatives also. They are known, $\displaystyle{\left({\sin{{\left({2}{x}\right)}}}\right)}'={2}{\cos{{\left({2}{x}\right)}}}$ (we use the chain rule for $\displaystyle{2}{x}$ here), $\displaystyle{\left({{\tanh}}^{{-{{1}}}}{\left({z}\right)}\right)}'=\frac{{1}}{{{1}-{{z}}^{{2}}}}.$

This way the result is $\displaystyle{y}'{\left({x}\right)}=\frac{{1}}{{{1}-{{\sin}}^{{2}}{\left({2}{x}\right)}}}\cdot{2}{\cos{{\left({2}{x}\right)}}},$

which is equal to $\displaystyle\frac{{{2}{\cos{{\left({2}{x}\right)}}}}}{{{{\cos}}^{{2}}{\left({2}{x}\right)}}}=\frac{{2}}{{{\cos{{\left({2}{x}\right)}}}}}={2}{\sec{{\left({2}{x}\right)}}}.$

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Saturday, September 12, 2009

$\displaystyle{y}={{\tanh}}^{{-{{1}}}}{\left({\sin{{\left({2}{x}\right)}}}\right)}$ 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?

Saturday, September 12, 2009

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?

$\displaystyle{y}={{\tanh}}^{{-{{1}}}}{\left({\sin{{\left({2}{x}\right)}}}\right)}$ 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?