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

Find the derivative of ((cos(x)-1)/(sin(x)))^(1/2)

Answer

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

Step-by-step explanation

Problem

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

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

3. See formulas

Time to solve it:

~ 0.8 seconds