`y = 2xsinh^-1(2x) - sqrt(1+4x^2)` Find the derivative of the function

Wednesday, December 8, 2010

$\displaystyle{y}={2}{x}{{\sinh}}^{{-{{1}}}}{\left({2}{x}\right)}-\sqrt{{{1}+{4}{{x}}^{{2}}}}$ Find the derivative of the function

The expression of this function includes difference, product and two table functions (except the polynomials), $\displaystyle{{\sinh}}^{{-{1}}}{\left({z}\right)}$ and $\displaystyle\sqrt{{{z}}}.$

The difference rule is $\displaystyle{\left({u}-{v}\right)}'={u}'-{v}',$ the product rule is $\displaystyle{\left({u}{v}\right)}'={u}'{v}+{u}{v}',$ the chain rule is $\displaystyle{\left({u}{\left({v}{\left({x}\right)}\right)}\right)}'={u}'{\left({v}{\left({x}\right)}\right)}\cdot{v}'{\left({x}\right)}.$

The derivative of $\displaystyle{{\sinh}}^{{-{1}}}{\left({z}\right)}$ is $\displaystyle\frac{{1}}{\sqrt{{{1}+{{z}}^{{2}}}}},$ the derivative of $\displaystyle\sqrt{{{z}}}$ is $\displaystyle\frac{{1}}{{{2}\sqrt{{{z}}}}}.$

These rules together give us

$\displaystyle{y}'={2}{{\sinh}}^{{-{{1}}}}{\left({2}{x}\right)}+{2}{x}\frac{{{\left({2}{x}\right)}'}}{\sqrt{{{1}+{{\left({2}{x}\right)}}^{{2}}}}}-\frac{{{\left({4}{{x}}^{{2}}\right)}'}}{{{2}\sqrt{{{1}+{{\left({2}{x}\right)}}^{{2}}}}}}=$

$\displaystyle={2}{{\sinh}}^{{-{{1}}}}{\left({2}{x}\right)}+{2}{x}\frac{{{2}}}{\sqrt{{{1}+{4}{{x}}^{{2}}}}}-\frac{{{8}{x}}}{{{2}\sqrt{{{1}+{4}{{x}}^{{2}}}}}}=$

$\displaystyle={2}{{\sinh}}^{{-{{1}}}}{\left({2}{x}\right)}.$

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Wednesday, December 8, 2010

$\displaystyle{y}={2}{x}{{\sinh}}^{{-{{1}}}}{\left({2}{x}\right)}-\sqrt{{{1}+{4}{{x}}^{{2}}}}$ 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?

Wednesday, December 8, 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?

$\displaystyle{y}={2}{x}{{\sinh}}^{{-{{1}}}}{\left({2}{x}\right)}-\sqrt{{{1}+{4}{{x}}^{{2}}}}$ 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?