# Find the derivative of ln((x/y)^0.5)

## \frac{d}{dx}\left(\ln\left(\sqrt{\frac{x}{y}}\right)\right)

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$\frac{\frac{1}{2}}{x}$

## Step by step solution

Problem

$\frac{d}{dx}\left(\ln\left(\sqrt{\frac{x}{y}}\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

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

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

$\frac{1}{2}\cdot\frac{1}{\sqrt{\frac{x}{y}}}\left(\frac{x}{y}\right)^{-\frac{1}{2}}\cdot\frac{d}{dx}\left(\frac{x}{y}\right)$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{1}{2}\cdot\frac{1}{\sqrt{\frac{x}{y}}}\left(\frac{x}{y}\right)^{-\frac{1}{2}}\cdot\frac{y\frac{d}{dx}\left(x\right)-x\frac{d}{dx}\left(y\right)}{y^2}$
4

The derivative of the constant function is equal to zero

$\frac{1}{2}\cdot\frac{1}{\sqrt{\frac{x}{y}}}\left(\frac{x}{y}\right)^{-\frac{1}{2}}\cdot\frac{0\left(-1\right)x+y\frac{d}{dx}\left(x\right)}{y^2}$
5

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

$\frac{1}{2}\cdot\frac{1}{\sqrt{\frac{x}{y}}}\left(\frac{x}{y}\right)^{-\frac{1}{2}}\cdot\frac{0+y\frac{d}{dx}\left(x\right)}{y^2}$
6

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

$\frac{1}{2}\cdot\frac{1}{\sqrt{\frac{x}{y}}}\cdot\frac{0+1y}{y^2}\left(\frac{x}{y}\right)^{-\frac{1}{2}}$
7

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

$\frac{1}{2}\cdot\frac{1}{\sqrt{\frac{x}{y}}}\cdot\frac{y}{y^2}\left(\frac{x}{y}\right)^{-\frac{1}{2}}$
8

Simplifying the fraction by $y$

$\frac{1}{2}\cdot\frac{1}{\sqrt{\frac{x}{y}}}\cdot\frac{1}{y}\left(\frac{x}{y}\right)^{-\frac{1}{2}}$
9

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}, where a=\frac{1}{2} and x=\sqrt{\frac{x}{y}} \frac{1}{y}\left(\frac{x}{y}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{x}{y}}} 10 The power of a quotient is equal to the quotient of the power of the numerator and denominator: \displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \frac{1}{y}\cdot\frac{x^{-\frac{1}{2}}}{y^{-\frac{1}{2}}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{x}{y}}} 11 The power of a quotient is equal to the quotient of the power of the numerator and denominator: \displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \frac{1}{y}\cdot\frac{x^{-\frac{1}{2}}}{y^{-\frac{1}{2}}}\cdot\frac{\frac{1}{2}}{\frac{\sqrt{x}}{\sqrt{y}}} 12 Simplifying the fraction \frac{1}{y}\cdot\frac{x^{-\frac{1}{2}}}{y^{-\frac{1}{2}}}\cdot\frac{1}{2}\cdot\frac{\sqrt{y}}{\sqrt{x}} 13 Multiplying the fraction and term \frac{1}{y}\cdot\frac{x^{-\frac{1}{2}}}{y^{-\frac{1}{2}}}\cdot\frac{\frac{1}{2}\sqrt{y}}{\sqrt{x}} 14 Multiplying fractions \frac{\frac{1}{2}\sqrt{y}}{\sqrt{x}}\cdot\frac{x^{-\frac{1}{2}}}{\sqrt{y}} 15 Multiplying fractions \frac{\frac{1}{2}x^{-\frac{1}{2}}\sqrt{y}}{\sqrt{x}\sqrt{y}} 16 Simplifying the fraction by \sqrt{y} \frac{\frac{1}{2}x^{-\frac{1}{2}}}{\sqrt{x}} 17 Simplifying the fraction by x \frac{1}{2}x^{\left(-\frac{1}{2}-\frac{1}{2}\right)} 18 Subtract the values -\frac{1}{2} and -\frac{1}{2} \frac{1}{2}x^{-1} 19 Applying the property of exponents, \displaystyle a^{-n}=\frac{1}{a^n}, where n is a number \frac{1}{2}\cdot\frac{1}{x} 20 Apply the formula: a\frac{1}{x}$$=\frac{a}{x}$, where $a=\frac{1}{2}$

$\frac{\frac{1}{2}}{x}$

$\frac{\frac{1}{2}}{x}$

### Main topic:

Differential calculus

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