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Step-by-step Solution

Find the derivative using the quotient rule $\frac{d}{dx}\left(\frac{2x^3\cot\left(x^2\right)}{1+e^{2x}}\right)$

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Answer

$\frac{2\left(3x^{2}\cot\left(x^2\right)-2\csc\left(x^2\right)^2x^{4}\right)}{1+e^{2x}}$

Step-by-step explanation

Problem to solve:

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

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

The derivative of the constant function ($b$) is equal to zero

$\frac{\frac{d}{dx}\left(2x^3\cot\left(x^2\right)\right)\left(1+e^{2x}\right)+0x^3\cot\left(x^2\right)}{\left(1+e^{2x}\right)^2}$

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Answer

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

Main topic:

Quotient rule of differentiation

Used formulas:

6. See formulas

Time to solve it:

~ 0.9 seconds