# Step-by-step Solution

## Derive the function $arcsin\left(\cos\left(\frac{x}{3}\right)\right)$ with respect to x

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## Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(arcsin\left(\cos\left(\frac{x}{3}\right)\right)\right)$
1

Taking the derivative of arcsine

$\frac{1}{\sqrt{1-\cos\left(\frac{x}{3}\right)^2}}\cdot\frac{d}{dx}\left(\cos\left(\frac{x}{3}\right)\right)$
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The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

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

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$\frac{d}{dx}\left(arcsin\left(\cos\left(\frac{x}{3}\right)\right)\right)$

### Main topic:

Differential calculus

~ 0.89 seconds