# Find the derivative of arccos(sin(x))

## \frac{d}{dx}\left(arccos\left(\sin\left(x\right)\right)\right)

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ln
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sin
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asin
acos
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sinh
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asinh
acosh
atanh
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## Step by step solution

Problem

$\frac{d}{dx}\left(arccos\left(\sin\left(x\right)\right)\right)$
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Taking the derivative of arccosine

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

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\cos\left(x\right)\frac{-1}{\sqrt{1-\sin\left(x\right)^2}}$
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Applying the trigonometric identity: $1-\sin\left(\theta\right)^2=\cos\left(\theta\right)^2$

$\cos\left(x\right)\frac{-1}{\sqrt{\cos\left(x\right)^2}}$
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Applying the power of a power property

$\cos\left(x\right)\frac{-1}{\cos\left(x\right)}$
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Multiplying the fraction and term

$\frac{-\cos\left(x\right)}{\cos\left(x\right)}$
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Simplifying the fraction by $\cos\left(x\right)$

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### Main topic:

Differential calculus

0.25 seconds

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