# Derive the function sin(2(1-1ln(x))/x) with respect to x

## \frac{d}{dx}\left(\sin\left(2\frac{1-\ln\left(x\right)}{x}\right)\right)

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

## Step by step solution

Problem

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

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

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

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

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

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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$2\cos\left(2\frac{1-\ln\left(x\right)}{x}\right)\frac{x\frac{d}{dx}\left(1-\ln\left(x\right)\right)-\frac{d}{dx}\left(x\right)\left(1-\ln\left(x\right)\right)}{x^2}$
4

The derivative of the linear function is equal to $1$

$2\cos\left(2\frac{1-\ln\left(x\right)}{x}\right)\frac{1\left(-1\right)\left(1-\ln\left(x\right)\right)+x\frac{d}{dx}\left(1-\ln\left(x\right)\right)}{x^2}$
5

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

$2\cos\left(2\frac{1-\ln\left(x\right)}{x}\right)\frac{1\left(-1\right)\left(1-\ln\left(x\right)\right)+x\left(\frac{d}{dx}\left(-\ln\left(x\right)\right)+\frac{d}{dx}\left(1\right)\right)}{x^2}$
6

The derivative of the constant function is equal to zero

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

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

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

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

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

The derivative of the linear function is equal to $1$

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

Multiply $1$ times $-1$

$2\cos\left(2\frac{1-\ln\left(x\right)}{x}\right)\frac{x\left(0-\frac{1}{x}\right)-\left(1-\ln\left(x\right)\right)}{x^2}$
11

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

$2\cos\left(2\frac{1-\ln\left(x\right)}{x}\right)\frac{-x\frac{1}{x}-\left(1-\ln\left(x\right)\right)}{x^2}$
12

Applying an identity of double-angle cosine

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

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

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### Main topic:

Differential calculus

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