`f(t) = t^(3/2)log_2sqrt(t+1)` Find the derivative of the function

Monday, January 11, 2016

$\displaystyle{f{{\left({t}\right)}}}={{t}}^{{\frac{{3}}{{2}}}}{\log}_{{2}}\sqrt{{{t}+{1}}}$ Find the derivative of the function

We shall use:

Product rule

$\displaystyle{\left({u}\cdot{v}\right)}'={u}'\cdot{v}+{u}\cdot{v}'$

Chain rule

$\displaystyle{\left({u}{\left({v}\right)}\right)}'={u}'{\left({v}\right)}\cdot{v}'$

$\displaystyle{u}$ and $\displaystyle{v}$ are both functions of arbitrary variable over which we are differentiating.

First we apply product rule.

$\displaystyle{f{'}}{\left({t}\right)}={\left({{t}}^{{\frac{{3}}{{2}}}}\right)}'{\log}_{{2}}\sqrt{{{t}+{1}}}+{{t}}^{{\frac{{3}}{{2}}}}{\left({\log}_{{2}}\sqrt{{{t}+{1}}}\right)}'=$

Now we apply the chain rule on the second term.

$\displaystyle\frac{{3}}{{2}}{{t}}^{{\frac{{1}}{{2}}}}{\log}_{{2}}\sqrt{{{t}+{1}}}+{{t}}^{{\frac{{3}}{{2}}}}\frac{{1}}{{\sqrt{{{t}+{1}}}{\ln{{2}}}}}\cdot\frac{{1}}{{{2}\sqrt{{{t}+{1}}}}}=$

Now we simplify the obtained derivative to get the final result.

$\displaystyle\frac{{3}}{{2}}\sqrt{{{t}}}{\log}_{{2}}\sqrt{{{t}+{1}}}+\frac{{{t}}^{{\frac{{3}}{{2}}}}}{{{2}{\left({t}+{1}\right)}{\ln{{2}}}}}$

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Monday, January 11, 2016

$\displaystyle{f{{\left({t}\right)}}}={{t}}^{{\frac{{3}}{{2}}}}{\log}_{{2}}\sqrt{{{t}+{1}}}$ 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?

Monday, January 11, 2016

Find the derivative of the function

No comments:

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?

$\displaystyle{f{{\left({t}\right)}}}={{t}}^{{\frac{{3}}{{2}}}}{\log}_{{2}}\sqrt{{{t}+{1}}}$ 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?