# Find the derivative of -6/((5x-1)^(1/3))

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

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

## Step by step solution

Problem

$\frac{d}{dx}\left(\frac{-6}{\left(5x-1\right)^{\frac{1}{3}}}\right)$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

The derivative of the constant function is equal to zero

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

Any expression multiplied by $0$ is equal to $0$

$\frac{6\frac{d}{dx}\left(\sqrt[3]{5x-1}\right)+0}{\left(\sqrt[3]{5x-1}\right)^2}$
4

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

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

The derivative of a sum of two functions is the sum of the derivatives of each function

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

The derivative of the constant function is equal to zero

$\frac{6\cdot \frac{1}{3}\left(5x-1\right)^{-\frac{2}{3}}\left(0+\frac{d}{dx}\left(5x\right)\right)+0}{\left(\sqrt[3]{5x-1}\right)^2}$
7

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{6\cdot \frac{1}{3}\left(5x-1\right)^{-\frac{2}{3}}\left(0+5\frac{d}{dx}\left(x\right)\right)+0}{\left(\sqrt[3]{5x-1}\right)^2}$
8

The derivative of the linear function is equal to $1$

$\frac{6\cdot \left(0+1\cdot 5\right)\cdot \frac{1}{3}\left(5x-1\right)^{-\frac{2}{3}}+0}{\left(\sqrt[3]{5x-1}\right)^2}$
9

Multiply $5$ times $1$

$\frac{\left(0+5\right)\cdot 2\left(5x-1\right)^{-\frac{2}{3}}+0}{\left(\sqrt[3]{5x-1}\right)^2}$
10

Add the values $5$ and $0$

$\frac{5\cdot 2\left(5x-1\right)^{-\frac{2}{3}}+0}{\left(\sqrt[3]{5x-1}\right)^2}$
11

Multiply $2$ times $5$

$\frac{10\left(5x-1\right)^{-\frac{2}{3}}+0}{\left(\sqrt[3]{5x-1}\right)^2}$
12

$x+0=x$, where $x$ is any expression

$\frac{10\left(5x-1\right)^{-\frac{2}{3}}}{\left(\sqrt[3]{5x-1}\right)^2}$
13

Applying the power of a power property

$\frac{10\left(5x-1\right)^{-\frac{2}{3}}}{\sqrt[3]{\left(5x-1\right)^{2}}}$
14

Simplifying the fraction by $5x-1$

$10\left(5x-1\right)^{\left(-\frac{2}{3}-\frac{2}{3}\right)}$
15

Subtract the values $-\frac{2}{3}$ and $-\frac{2}{3}$

$10\left(5x-1\right)^{-\frac{4}{3}}$
16

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$10\frac{1}{\sqrt[3]{\left(5x-1\right)^{4}}}$
17

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=10$ and $x=\sqrt[3]{\left(5x-1\right)^{4}}$

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

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

### Main topic:

Differential calculus

0.27 seconds

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