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Step-by-step Solution

Derive the function $\ln\left(\frac{\sqrt{2+x^2}}{x^2}\right)$ with respect to x

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Answer

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

Step-by-step explanation

Problem to solve:

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

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

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 0.94 seconds

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