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Step-by-step Solution

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

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Answer

$5\left(\frac{1-2x^3}{1+2x^3}\right)^{4}\left(\frac{-6x^{2}\left(1+2x^3\right)-6x^{2}\left(1-2x^3\right)}{\left(1+2x^3\right)^2}\right)$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\left(\frac{1-2x^3}{1+2x^3}\right)^5\right)$
1

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

$5\left(\frac{1-2x^3}{1+2x^3}\right)^{4}\cdot\frac{d}{dx}\left(\frac{1-2x^3}{1+2x^3}\right)$
2

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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 0.66 seconds