`f(t) = arcsin(t^2)` Find the derivative of the function

Friday, July 16, 2010

$\displaystyle{f{{\left({t}\right)}}}={\arcsin{{\left({{t}}^{{2}}\right)}}}$ Find the derivative of the function

The derivative of a function with respect to t is denoted as f'(t)

The given function: $\displaystyle{f{{\left({x}\right)}}}={\arcsin{{\left({{t}}^{{2}}\right)}}}$ is in a form of a inverse trigonometric function.

Using table of derivatives, we have the basic formula:

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

By comparison, we may let $\displaystyle{u}={{t}}^{{2}}$ then $\displaystyle\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}={2}{t}$ .

Applying the formula, we get:

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{{2}{t}}}{\sqrt{{{1}-{{\left({{t}}^{{2}}\right)}}^{{2}}}}}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{{2}{t}}}{\sqrt{{{1}-{{t}}^{{4}}}}}$ as the first derivative of $\displaystyle{f{{\left({x}\right)}}}={\arcsin{{\left({{t}}^{{2}}\right)}}}$

Note: $\displaystyle{{\left({{t}}^{{2}}\right)}}^{{2}}={{t}}^{{{2}\cdot{2}}}={{t}}^{{4}}$ based on the Law of Exponents: $\displaystyle{{\left({{x}}^{{m}}\right)}}^{{n}}={{x}}^{{{m}\cdot{n}}}$

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Friday, July 16, 2010

$\displaystyle{f{{\left({t}\right)}}}={\arcsin{{\left({{t}}^{{2}}\right)}}}$ 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, July 16, 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?

$\displaystyle{f{{\left({t}\right)}}}={\arcsin{{\left({{t}}^{{2}}\right)}}}$ 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?