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Step-by-step Solution

Find the derivative of $\frac{1}{2}\left(x\sqrt{4-x^2}+4arcsin\left(\frac{x}{2}\right)\right)$

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Answer

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

Step-by-step explanation

Problem to solve:

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

Solve the product $\frac{1}{2}\left(x\sqrt{4-x^2}+4arcsin\left(\frac{x}{2}\right)\right)$

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

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

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

9. See formulas

Time to solve it:

~ 0.82 seconds