`g(x) = arcsin(3x)/x` Find the derivative of the function

Wednesday, August 25, 2010

$\displaystyle{g{{\left({x}\right)}}}=\frac{{\arcsin{{\left({3}{x}\right)}}}}{{x}}$ Find the derivative of the function

The given function $\displaystyle{g{{\left({x}\right)}}}$ has a form $\displaystyle{u}\frac{{{x}}}{{v}}{\left({x}\right)},$ thus its derivative may be determined by the quotient rule: $\displaystyle{\left(\frac{{u}}{{v}}\right)}'=\frac{{{u}'{v}-{u}{v}'}}{{{v}}^{{2}}}.$

Also we need to know that the derivative of acrsine function is $\displaystyle\frac{{1}}{\sqrt{{{1}-{{x}}^{{2}}}}},$ and a bit of the chain rule.

The result of applying these rules is

$\displaystyle{\left(\frac{{\arcsin{{\left({3}{x}\right)}}}}{{x}}\right)}'=\frac{{{x}\cdot{\left({\arcsin{{\left({3}{x}\right)}}}\right)}'-{\arcsin{{\left({3}{x}\right)}}}}}{{{x}}^{{2}}}=\frac{{\frac{{{3}{x}}}{\sqrt{{{1}-{9}{{x}}^{{2}}}}}-{\arcsin{{\left({3}{x}\right)}}}}}{{{x}}^{{2}}},$

or it may be written as $\displaystyle\frac{{3}}{{{x}\sqrt{{{1}-{9}{{x}}^{{2}}}}}}-\frac{{\arcsin{{\left({3}{x}\right)}}}}{{{x}}^{{2}}}.$

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Wednesday, August 25, 2010

$\displaystyle{g{{\left({x}\right)}}}=\frac{{\arcsin{{\left({3}{x}\right)}}}}{{x}}$ 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, August 25, 2010

Find the derivative of the function

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

$\displaystyle{g{{\left({x}\right)}}}=\frac{{\arcsin{{\left({3}{x}\right)}}}}{{x}}$ 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?