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Step-by-step Solution

Find the derivative of $\ln\left(\frac{6\left(x-9\right)^3x^3}{\left(x-2\right)^{\left(\frac{1}{5}\right)}}\right)$

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Answer

$\frac{3\left(x-9\right)^{2}x^3+3\left(x-9\right)^3x^{2}}{\left(x-9\right)^3x^3}$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(ln\left(\frac{\left(x-9\right)^3\left(6x^3\right)}{\sqrt[5]{x-2}}\right)\right)$
1

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{\frac{6\left(x-9\right)^3x^3}{\sqrt[5]{x-2}}}\cdot\frac{d}{dx}\left(\frac{6\left(x-9\right)^3x^3}{\sqrt[5]{x-2}}\right)$
2

Simplifying the fraction

$\frac{\sqrt[5]{x-2}}{6\left(x-9\right)^3x^3}\cdot\frac{d}{dx}\left(\frac{6\left(x-9\right)^3x^3}{\sqrt[5]{x-2}}\right)$

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Answer

$\frac{3\left(x-9\right)^{2}x^3+3\left(x-9\right)^3x^{2}}{\left(x-9\right)^3x^3}$
$\frac{d}{dx}\left(ln\left(\frac{\left(x-9\right)^3\left(6x^3\right)}{\sqrt[5]{x-2}}\right)\right)$

Main topic:

Differential calculus

Related formulas:

8. See formulas

Time to solve it:

~ 0.23 seconds