`g(x) = 3arccos(x/2)` Find the derivative of the function

Friday, May 6, 2011

$\displaystyle{g{{\left({x}\right)}}}={3}{\arccos{{\left(\frac{{x}}{{2}}\right)}}}$ Find the derivative of the function

To take the derivative of the given function: $\displaystyle{g{{\left({x}\right)}}}={3}{\arccos{{\left(\frac{{x}}{{2}}\right)}}}$ ,

we can apply the basic property: $\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left[{c}\cdot{f{{\left({x}\right)}}}\right]}={c}\cdot\frac{{d}}{{{\left.{d}{x}\right.}}}{\left[{f{{\left({x}\right)}}}\right]}$ .

then $\displaystyle{g{'}}{\left({x}\right)}={3}\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}$

To solve for the $\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}$ , we consider the derivative formula of an inverse trigonometric function.

For the derivative of inverse "cosine" function, we follow:

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

To apply the formula with the given function, we let $\displaystyle{u}=\frac{{x}}{{2}}$ then $\displaystyle\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}=\frac{{1}}{{2}}$ .

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

Evaluate the exponent:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}=-\frac{{\frac{{1}}{{2}}}}{\sqrt{{{1}-\frac{{{x}}^{{2}}}{{4}}}}}$

Express the expression inside radical as one fraction:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}=-\frac{{\frac{{1}}{{2}}}}{\sqrt{{\frac{{{4}-{{x}}^{{2}}}}{{4}}}}}$

Apply the property of radicals: $\displaystyle\sqrt{{\frac{{a}}{{b}}}}=\frac{\sqrt{{{a}}}}{\sqrt{{{b}}}}$ at the bottom:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}=-\frac{{\frac{{1}}{{2}}}}{{{\left(\frac{\sqrt{{{4}-{{x}}^{{2}}}}}{\sqrt{{{4}}}}\right)}}}$

To simplify, flip the bottom to proceed to multiplication:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}=-{\left(\frac{{1}}{{2}}\right)}\cdot\frac{\sqrt{{{4}}}}{\sqrt{{{\left({4}-{{x}}^{{2}}\right)}}}}$

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}=-{\left(\frac{{1}}{{2}}\right)}\cdot\frac{{2}}{\sqrt{{{\left({4}-{{x}}^{{2}}\right)}}}}$

Multiply across:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}=-\frac{{2}}{{{2}\sqrt{{{4}-{{x}}^{{2}}}}}}$

Cancel out the common factor 2 from top and bottom:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}=-\frac{{1}}{\sqrt{{{4}-{{x}}^{{2}}}}}$

With $\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}=-\frac{{1}}{\sqrt{{{4}-{{x}}^{{2}}}}}$ , then

$\displaystyle{g{'}}{\left({x}\right)}={3}\cdot\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left(\frac{{x}}{{2}}\right)}}}\right)}$

becomes

$\displaystyle{g{'}}{\left({x}\right)}={3}\cdot-\frac{{1}}{\sqrt{{{4}-{{x}}^{{2}}}}}$

$\displaystyle{g{'}}{\left({x}\right)}=-\frac{{3}}{\sqrt{{{4}-{{x}}^{{2}}}}}$

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Friday, May 6, 2011

$\displaystyle{g{{\left({x}\right)}}}={3}{\arccos{{\left(\frac{{x}}{{2}}\right)}}}$ 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?

Friday, May 6, 2011

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?

$\displaystyle{g{{\left({x}\right)}}}={3}{\arccos{{\left(\frac{{x}}{{2}}\right)}}}$ 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?