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Step-by-step Solution

Derive the function ln((x^2+2*x)^0.5) with respect to x

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Answer

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

Step-by-step explanation

Problem to solve:

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

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

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

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 0.84 seconds

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