`f(x) =x9^x` Find the derivative of the function

Wednesday, September 3, 2014

$\displaystyle{f{{\left({x}\right)}}}={x}{{9}}^{{x}}$ Find the derivative of the function

We can apply the Product Rule for derivatives:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({u}\cdot{v}\right)}={u}'\cdot{v}+{u}\cdot{v}$ '.

With the given function: $\displaystyle{f{{\left({x}\right)}}}={x}\cdot{{9}}^{{x}}$ , we may let:

$\displaystyle{u}={x}$ then $\displaystyle{u}'={1}$

Using the formula for the derivative of an exponential function:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({{a}}^{{u}}\right)}={{a}}^{{u}}\cdot{\ln{{\left({a}\right)}}}\cdot\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}$ where $\displaystyle{a}\ne{1}$ .

Then $\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({{9}}^{{x}}\right)}={{9}}^{{x}}\cdot{\ln{{\left({9}\right)}}}={{9}}^{{x}}{\ln{{\left({9}\right)}}}$ where $\displaystyle{a}={9}$ and $\displaystyle{u}={x}$ .

Using Product Rule with the values:

$\displaystyle{u}={x}$ , $\displaystyle{u}'={1}$ , $\displaystyle{v}={{9}}^{{x}}$ and v $\displaystyle'={{9}}^{{x}}{\ln{{\left({9}\right)}}}$ ,we get:

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}\cdot{{9}}^{{x}}\right)}={1}\cdot{{9}}^{{x}}+{x}\cdot{{9}}^{{x}}{\ln{{\left({9}\right)}}}$

$\displaystyle{f{'}}{\left({x}\right)}={{9}}^{{x}}+{x}{{9}}^{{x}}{\ln{{\left({9}\right)}}}$

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Wednesday, September 3, 2014

$\displaystyle{f{{\left({x}\right)}}}={x}{{9}}^{{x}}$ 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?

Wednesday, September 3, 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?

$\displaystyle{f{{\left({x}\right)}}}={x}{{9}}^{{x}}$ 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?