`g(t) = log_2(t^2+7)^3` Find the derivative of the function

Saturday, August 30, 2014

$\displaystyle{g{{\left({t}\right)}}}={\log}_{{2}}{{\left({{t}}^{{2}}+{7}\right)}}^{{3}}$ Find the derivative of the function

$\displaystyle{g{{\left({t}\right)}}}={\log}_{{2}}{{\left({{t}}^{{2}}+{7}\right)}}^{{3}}$

Before taking the derivative of the function, apply the logarithm rule $\displaystyle{\log}_{{b}}{\left({{a}}^{{m}}\right)}={m}\cdot{\log}_{{b}}{\left({a}\right)}$ . So the function becomes:

$\displaystyle{g{{\left({t}\right)}}}={3}{\log}_{{2}}{\left({{t}}^{{2}}+{7}\right)}$

Take note that the derivative formula of logarithm is $\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[{\log}_{{b}}{\left({u}\right)}\right]}=\frac{{1}}{{{\ln{{\left({b}\right)}}}\cdot{u}}}\cdot\frac{{{d}{u}}}{{\left.{d}{x}}}$ .

So g'(t) will be:

$\displaystyle{g{'}}{\left({t}\right)}=\frac{{d}}{{\left.{d}{t}}}{\left[{3}{\log}_{{2}}{\left({{t}}^{{2}}+{7}\right)}\right]}$

$\displaystyle{g{'}}{\left({t}\right)}={3}\frac{{d}}{{\left.{d}{t}}}{\left[{\log}_{{2}}{\left({{t}}^{{2}}+{7}\right)}\right]}$

$\displaystyle{g{'}}{\left({t}\right)}={3}\cdot\frac{{1}}{{{\ln{{\left({2}\right)}}}\cdot{\left({{t}}^{{2}}+{7}\right)}}}\cdot\frac{{d}}{{\left.{d}{t}}}{\left({{t}}^{{2}}+{7}\right)}$

$\displaystyle{g{'}}{\left({t}\right)}={3}\cdot\frac{{1}}{{{\ln{{\left({2}\right)}}}\cdot{\left({{t}}^{{2}}+{7}\right)}}}\cdot{2}{t}$

$\displaystyle{g{'}}{\left({t}\right)}=\frac{{{3}\cdot{2}{t}}}{{{\ln{{\left({2}\right)}}}\cdot{\left({{t}}^{{2}}+{7}\right)}}}$

$\displaystyle{g{'}}{\left({t}\right)}=\frac{{{6}{t}}}{{{\left({{t}}^{{2}}+{7}\right)}{\ln{{\left({2}\right)}}}}}$

Therefore, the derivative of the function is $\displaystyle{g{'}}{\left({t}\right)}=\frac{{{6}{t}}}{{{\left({{t}}^{{2}}+{7}\right)}{\ln{{\left({2}\right)}}}}}$ .

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Saturday, August 30, 2014

$\displaystyle{g{{\left({t}\right)}}}={\log}_{{2}}{{\left({{t}}^{{2}}+{7}\right)}}^{{3}}$ 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?

Saturday, August 30, 2014

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?

$\displaystyle{g{{\left({t}\right)}}}={\log}_{{2}}{{\left({{t}}^{{2}}+{7}\right)}}^{{3}}$ 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?