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

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## Answer

$\frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)}{\frac{1}{4}\left(\sin\left(x\right)-\cos\left(x\right)\right)\ln\left(\sin\left(x\right)-\cos\left(x\right)\right)^2}$

## Step by step solution

Problem

$\frac{d}{dx}\left(\frac{3}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)}\right)$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

The derivative of the constant function is equal to zero

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

Any expression multiplied by $0$ is equal to $0$

$\frac{0-3\frac{d}{dx}\left(\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)\right)}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
4

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{0-3\frac{d}{dx}\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)\left(\frac{1}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}\right)}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
5

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

$\frac{0-3\cdot \frac{1}{2}\left(\frac{1}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}\right)\frac{d}{dx}\left(\sin\left(x\right)-\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
6

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{0-3\cdot \frac{1}{2}\left(\frac{1}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}\right)\left(\frac{d}{dx}\left(-\cos\left(x\right)\right)+\frac{d}{dx}\left(\sin\left(x\right)\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
7

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{0-3\cdot \frac{1}{2}\left(\frac{1}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}\right)\left(\frac{d}{dx}\left(\sin\left(x\right)\right)-\frac{d}{dx}\left(\cos\left(x\right)\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
8

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{0-3\cdot \frac{1}{2}\left(\frac{1}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}\right)\left(\cos\left(x\right)-\frac{d}{dx}\left(\cos\left(x\right)\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
9

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 $f'(x) = -\sin(x)\cdot D_x(x)$

$\frac{0-3\cdot \frac{1}{2}\left(\frac{1}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}\right)\left(\cos\left(x\right)-1\left(-1\right)\sin\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
10

Multiply $-1$ times $-1$

$\frac{0-\frac{3}{2}\cdot\frac{1}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}\left(1\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
11

$x+0=x$, where $x$ is any expression

$\frac{-\frac{3}{2}\cdot\frac{1}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2}$
12

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}, where a=-\frac{3}{2} and x=\sqrt{\sin\left(x\right)-\cos\left(x\right)} \frac{\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}\cdot\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} 13 Multiplying the fraction and term \frac{\frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-\frac{1}{2}}}{\sqrt{\sin\left(x\right)-\cos\left(x\right)}}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2} 14 Simplifying the fraction by \sin\left(x\right)-\cos\left(x\right) \frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{\left(-\frac{1}{2}-\frac{1}{2}\right)}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2} 15 Subtract the values -\frac{1}{2} and -\frac{1}{2} \frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-1}}{\ln\left(\sqrt{\sin\left(x\right)-\cos\left(x\right)}\right)^2} 16 Using the power rule of logarithms: \log_a(x^n)=n\cdot\log_a(x) \frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-1}}{\left(\frac{1}{2}\ln\left(\sin\left(x\right)-\cos\left(x\right)\right)\right)^2} 17 The power of a product is equal to the product of it's factors raised to the same power \frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)^{-1}}{\frac{1}{4}\ln\left(\sin\left(x\right)-\cos\left(x\right)\right)^2} 18 Applying the property of exponents, \displaystyle a^{-n}=\frac{1}{a^n}, where n is a number \frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\frac{1}{\sin\left(x\right)-\cos\left(x\right)}\right)}{\frac{1}{4}\ln\left(\sin\left(x\right)-\cos\left(x\right)\right)^2} 19 Apply the formula: a\frac{1}{x}$$=\frac{a}{x}$, where $a=-\frac{3}{2}$ and $x=\sin\left(x\right)-\cos\left(x\right)$

$\frac{\left(\sin\left(x\right)+\cos\left(x\right)\right)\frac{-\frac{3}{2}}{\sin\left(x\right)-\cos\left(x\right)}}{\frac{1}{4}\ln\left(\sin\left(x\right)-\cos\left(x\right)\right)^2}$
20

Multiplying the fraction and term

$\frac{\frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)}{\sin\left(x\right)-\cos\left(x\right)}}{\frac{1}{4}\ln\left(\sin\left(x\right)-\cos\left(x\right)\right)^2}$
21

Simplifying the fraction

$\frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)}{\frac{1}{4}\left(\sin\left(x\right)-\cos\left(x\right)\right)\ln\left(\sin\left(x\right)-\cos\left(x\right)\right)^2}$

## Answer

$\frac{-\frac{3}{2}\left(\sin\left(x\right)+\cos\left(x\right)\right)}{\frac{1}{4}\left(\sin\left(x\right)-\cos\left(x\right)\right)\ln\left(\sin\left(x\right)-\cos\left(x\right)\right)^2}$

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### Main topic:

Differential calculus

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