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Step-by-step Solution

Derive the function 3/(ln((sin(x)-1*cos(x))^0.5) with respect to x

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Answer

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

Step-by-step explanation

Problem to solve:

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

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

The derivative of the constant function is equal to zero

$\frac{0\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)-3\frac{d}{dx}\left(\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)\right)}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$

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Answer

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

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 0.97 seconds

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