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Step-by-step Solution

Find the derivative using the product rule (d/dx)(2/3arccos((x^3+3*x)^0.5))

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Answer

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

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\frac{2}{3}\cdot arccos\left(\sqrt{x^3+3x}\right)\right)$
1

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

$\frac{2}{3}\cdot\frac{d}{dx}\left(arccos\left(\sqrt{x^3+3x}\right)\right)$
2

Taking the derivative of arccosine

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

4. See formulas

Time to solve it:

~ 1.39 seconds