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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Answer

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

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 $a=\frac{1}{2}$

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

Answer

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

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.38 seconds

Views:

109