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

Find the derivative of 1/2(x(4-1x^2)^0.5+4arcsin(x/2))

Answer

$\frac{1}{2}\cdot\frac{2}{\sqrt{\frac{-x^2}{4}+1}}+\frac{-\frac{1}{2}x^2}{\sqrt{4-x^2}}+\frac{1}{2}\sqrt{4-x^2}$

Step-by-step explanation

Problem

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\frac{1}{2}$ and $g=4arcsin\left(\frac{x}{2}\right)+\sqrt{4-x^2}x$

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

6. See formulas

Time to solve it:

~ 2.63 seconds