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Step-by-step Solution

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

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Answer

$10\left(5x-1\right)^{-\frac{4}{3}}$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\frac{-6}{\left(5x-1\right)^{\frac{1}{3}}}\right)$
1

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

The derivative of the constant function is equal to zero

$\frac{0\sqrt[3]{5x-1}+6\frac{d}{dx}\left(\sqrt[3]{5x-1}\right)}{\left(\sqrt[3]{5x-1}\right)^2}$

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Answer

$10\left(5x-1\right)^{-\frac{4}{3}}$
$\frac{d}{dx}\left(\frac{-6}{\left(5x-1\right)^{\frac{1}{3}}}\right)$

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 1.6 seconds