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

Tuesday, June 11, 2013

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

$\displaystyle{f{{\left({t}\right)}}}=\frac{{{3}}^{{{2}{t}}}}{{t}}$

To take the derivative of this function, use the quotient rule $\displaystyle{\left(\frac{{u}}{{v}}\right)}'=\frac{{{v}\cdot{u}'-{u}\cdot{v}'}}{{{v}}^{{2}}}$ .

Applying that, f'(t) will be:

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{{t}\cdot{\left({{3}}^{{{2}{t}}}\right)}'-{{3}}^{{{2}{t}}}\cdot{\left({t}\right)}'}}{{{t}}^{{2}}}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{{t}\cdot{\left({{3}}^{{{2}{t}}}\right)}'-{{3}}^{{{2}{t}}}\cdot{1}}}{{{t}}^{{2}}}$

Take note that the derivative formula of an exponential function is $\displaystyle{\left({{a}}^{{u}}\right)}'={\ln{{\left({a}\right)}}}\cdot{{a}}^{{u}}\cdot{u}'$ .

So the derivative of $\displaystyle{{3}}^{{{2}{t}}}$ is:

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{{t}\cdot{\ln{{\left({3}\right)}}}\cdot{{3}}^{{{2}{t}}}\cdot{\left({2}{t}\right)}'-{{3}}^{{{2}{t}}}\cdot{1}}}{{{t}}^{{2}}}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{{t}\cdot{\ln{{\left({3}\right)}}}\cdot{{3}}^{{{2}{t}}}\cdot{2}-{{3}}^{{{2}{t}}}\cdot{1}}}{{{t}}^{{2}}}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{{2}{t}{\ln{{\left({3}\right)}}}\cdot{{3}}^{{{2}{t}}}-{{3}}^{{{2}{t}}}}}{{{t}}^{{2}}}$

$\displaystyle{f{'}}{\left({t}\right)}=\frac{{{{3}}^{{{2}{t}}}{\left({2}{t}{\ln{{\left({3}\right)}}}-{1}\right)}}}{{{t}}^{{2}}}$

Therefore, the derivative of the function is $\displaystyle{f{'}}{\left({t}\right)}=\frac{{{{3}}^{{{2}{t}}}{\left({2}{t}{\ln{{\left({3}\right)}}}-{1}\right)}}}{{{t}}^{{2}}}$ .

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Tuesday, June 11, 2013

$\displaystyle{f{{\left({t}\right)}}}=\frac{{{3}}^{{{2}{t}}}}{{t}}$ 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?

Tuesday, June 11, 2013

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?

$\displaystyle{f{{\left({t}\right)}}}=\frac{{{3}}^{{{2}{t}}}}{{t}}$ 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?