`f(x) = arcsin(x) + arccos(x)` Find the derivative of the function

Thursday, January 28, 2010

$\displaystyle{f{{\left({x}\right)}}}={\arcsin{{\left({x}\right)}}}+{\arccos{{\left({x}\right)}}}$ Find the derivative of the function

Recall that the derivative of a function f at a point x is denoted as f'(x).

The given function: $\displaystyle{f{{\left({x}\right)}}}={\arcsin{{\left({x}\right)}}}+{\arccos{{\left({x}\right)}}}$ has inverse trigonometric terms.

There are basic formulas for the derivative of inverse trigonometric functions:

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

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

Applying the formula in the given function:

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arcsin{{\left({x}\right)}}}\right)}+\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\arccos{{\left({x}\right)}}}\right)}$

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

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{1}}{\sqrt{{{1}-{{x}}^{{2}}}}}-\frac{{1}}{\sqrt{{{1}-{{x}}^{{2}}}}}$

$\displaystyle{f{'}}{\left({x}\right)}={0}$

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Thursday, January 28, 2010

$\displaystyle{f{{\left({x}\right)}}}={\arcsin{{\left({x}\right)}}}+{\arccos{{\left({x}\right)}}}$ 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?

Thursday, January 28, 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?

$\displaystyle{f{{\left({x}\right)}}}={\arcsin{{\left({x}\right)}}}+{\arccos{{\left({x}\right)}}}$ 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?