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# Find the derivative of ln(((2+x^2)^0.5)/(x^2))

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## Answer

$\frac{\frac{1}{\sqrt{x^2+2}}x^{3}-2x\sqrt{x^2+2}}{\sqrt{x^2+2}x^{2}}$

## Step by step solution

Problem

$\frac{d}{dx}\left(\ln\left(\frac{\sqrt{2+x^2}}{x^2}\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

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

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

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

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

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

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

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

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

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

The derivative of the constant function is equal to zero

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

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

$\frac{\frac{1}{2}x^2\left(2x+0\right)\left(x^2+2\right)^{-\frac{1}{2}}-1\cdot 2x\sqrt{x^2+2}}{\left(x^2\right)^2}\cdot\frac{1}{\frac{\sqrt{x^2+2}}{x^2}}$
8

Multiply $2$ times $-1$

$\frac{\frac{1}{2}x^2\left(2x+0\right)\left(x^2+2\right)^{-\frac{1}{2}}-2x\sqrt{x^2+2}}{\left(x^2\right)^2}\cdot\frac{1}{\frac{\sqrt{x^2+2}}{x^2}}$
9

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

$\frac{\frac{1}{2}\cdot 2xx^2\left(x^2+2\right)^{-\frac{1}{2}}-2x\sqrt{x^2+2}}{\left(x^2\right)^2}\cdot\frac{1}{\frac{\sqrt{x^2+2}}{x^2}}$
10

Multiply $2$ times $\frac{1}{2}$

$\frac{1xx^2\left(x^2+2\right)^{-\frac{1}{2}}-2x\sqrt{x^2+2}}{\left(x^2\right)^2}\cdot\frac{1}{\frac{\sqrt{x^2+2}}{x^2}}$
11

Any expression multiplied by $1$ is equal to itself

$\frac{xx^2\left(x^2+2\right)^{-\frac{1}{2}}-2x\sqrt{x^2+2}}{\left(x^2\right)^2}\cdot\frac{1}{\frac{\sqrt{x^2+2}}{x^2}}$
12

Applying the power of a power property

$\frac{xx^2\left(x^2+2\right)^{-\frac{1}{2}}-2x\sqrt{x^2+2}}{x^{4}}\cdot\frac{1}{\frac{\sqrt{x^2+2}}{x^2}}$
13

When multiplying exponents with same base you can add the exponents

$\frac{\left(x^2+2\right)^{-\frac{1}{2}}x^{3}-2x\sqrt{x^2+2}}{x^{4}}\cdot\frac{1}{\frac{\sqrt{x^2+2}}{x^2}}$
14

Simplifying the fraction

$\frac{\left(x^2+2\right)^{-\frac{1}{2}}x^{3}-2x\sqrt{x^2+2}}{x^{4}}\cdot\frac{x^2}{\sqrt{x^2+2}}$
15

Multiplying fractions

$\frac{\left(\left(x^2+2\right)^{-\frac{1}{2}}x^{3}-2x\sqrt{x^2+2}\right)x^2}{x^{4}\sqrt{x^2+2}}$
16

Simplifying the fraction by $x$

$\frac{\left(\left(x^2+2\right)^{-\frac{1}{2}}x^{3}-2x\sqrt{x^2+2}\right)x^{\left(2-4\right)}}{\sqrt{x^2+2}}$
17

Subtract the values $2$ and $-4$

$\frac{\left(\left(x^2+2\right)^{-\frac{1}{2}}x^{3}-2x\sqrt{x^2+2}\right)x^{-2}}{\sqrt{x^2+2}}$
18

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{\left(\frac{1}{\sqrt{x^2+2}}x^{3}-2x\sqrt{x^2+2}\right)\frac{1}{x^{2}}}{\sqrt{x^2+2}}$
19

Multiplying the fraction and term

$\frac{\frac{\frac{1}{\sqrt{x^2+2}}x^{3}-2x\sqrt{x^2+2}}{x^{2}}}{\sqrt{x^2+2}}$
20

Simplifying the fraction

$\frac{\frac{1}{\sqrt{x^2+2}}x^{3}-2x\sqrt{x^2+2}}{\sqrt{x^2+2}x^{2}}$

## Answer

$\frac{\frac{1}{\sqrt{x^2+2}}x^{3}-2x\sqrt{x^2+2}}{\sqrt{x^2+2}x^{2}}$

### Main topic:

Differential calculus

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