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Step-by-step Solution

Derive the function ln(arccos((1/(x^0.5)))) with respect to x

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Answer

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

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\ln\left(arccos\left(\frac{1}{\sqrt{x}}\right)\right)\right)$
1

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

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

Taking the derivative of arccosine

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

9. See formulas

Time to solve it:

~ 1.43 seconds