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\frac{d}{dx}\left(\ln\left(x\sqrt{a+x}\right)\right)

Find the derivative of ln(x(a+x)^0.5)

Answer

$\frac{a+\frac{3}{2}x}{x^2+a\cdot x}$

Step-by-step explanation

Problem

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

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

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Answer

$\frac{a+\frac{3}{2}x}{x^2+a\cdot x}$

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$\frac{d}{dx}\left(\ln\left(x\sqrt{a+x}\right)\right)$

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 0.43 seconds