# Find the derivative of ((3x-1)/(x^2+3))^2

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

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

## Step by step solution

Problem

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

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

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

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

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

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

The derivative of the constant function is equal to zero

$2\frac{3x-1}{3+x^2}\cdot\frac{\left(3+x^2\right)\left(0+\frac{d}{dx}\left(3x\right)\right)-\left(3x-1\right)\left(0+\frac{d}{dx}\left(x^2\right)\right)}{\left(3+x^2\right)^2}$
5

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$2\frac{3x-1}{3+x^2}\cdot\frac{\left(3+x^2\right)\left(0+3\frac{d}{dx}\left(x\right)\right)-\left(3x-1\right)\left(0+\frac{d}{dx}\left(x^2\right)\right)}{\left(3+x^2\right)^2}$
6

The derivative of the linear function is equal to $1$

$2\frac{3x-1}{3+x^2}\cdot\frac{\left(0+1\cdot 3\right)\left(3+x^2\right)-\left(3x-1\right)\left(0+\frac{d}{dx}\left(x^2\right)\right)}{\left(3+x^2\right)^2}$
7

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

$2\frac{\left(0+1\cdot 3\right)\left(3+x^2\right)-\left(0+2x\right)\left(3x-1\right)}{\left(3+x^2\right)^2}\cdot\frac{3x-1}{3+x^2}$
8

Multiply $3$ times $1$

$2\frac{\left(0+3\right)\left(3+x^2\right)-\left(0+2x\right)\left(3x-1\right)}{\left(3+x^2\right)^2}\cdot\frac{3x-1}{3+x^2}$
9

Add the values $3$ and $0$

$2\frac{3\left(3+x^2\right)-\left(0+2x\right)\left(3x-1\right)}{\left(3+x^2\right)^2}\cdot\frac{3x-1}{3+x^2}$
10

$x+0=x$, where $x$ is any expression

$2\frac{3\left(3+x^2\right)-1\cdot 2x\left(3x-1\right)}{\left(3+x^2\right)^2}\cdot\frac{3x-1}{3+x^2}$
11

Multiply $2$ times $-1$

$2\frac{3\left(3+x^2\right)-2x\left(3x-1\right)}{\left(3+x^2\right)^2}\cdot\frac{3x-1}{3+x^2}$
12

Multiplying fractions

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

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

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### Main topic:

Differential calculus

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