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Step-by-step Solution

Find the derivative $\frac{d}{dx}\left(\ln\left(xy^2-y\right)+\sqrt{3x-y}\right)$ using the sum rule

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Answer

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

Step-by-step explanation

Problem to solve:

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

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

$\frac{d}{dx}\left(\ln\left(xy^2-y\right)\right)+\frac{d}{dx}\left(\sqrt{3x-y}\right)$
2

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

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

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Answer

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

Main topic:

Sum rule

Used formulas:

5. See formulas

Time to solve it:

~ 0.65 seconds