# Step-by-step Solution

## Find the derivative of $\left(\frac{3x-1}{x^2+3}\right)^2$

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### Videos

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

## Step-by-step explanation

Problem to solve:

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

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

$2\left(\frac{3x-1}{x^2+3}\right)\frac{d}{dx}\left(\frac{3x-1}{x^2+3}\right)$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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

### Main topic:

Differential calculus

~ 0.64 seconds