1
Solved example of basic differentiation rules
$\frac{d}{dx}\left(\frac{x^2+3x+1}{x^2+2x+2}\right)^2$
2
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
$\frac{d}{dx}\left(\frac{\left(x^2+3x+1\right)^2}{\left(x^2+2x+2\right)^2}\right)$
Intermediate steps
Apply the quotient rule for differentiation, 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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(\left(x^2+2x+2\right)^2\right)^2}$
Simplify $\left(\left(x^2+2x+2\right)^2\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
3
Apply the quotient rule for differentiation, 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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(\left(x^2+2x+2\right)^2\right)^2}$
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Intermediate steps
Simplify $\left(\left(x^2+2x+2\right)^2\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{2\cdot 2}}$
Simplify $\left(\left(x^2+2x+2\right)^2\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{2\cdot 2}}$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
4
Simplify $\left(\left(x^2+2x+2\right)^2\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2$
$\frac{\frac{d}{dx}\left(\left(x^2+3x+1\right)^2\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
Explain more
5
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{2\left(x^2+3x+1\right)\frac{d}{dx}\left(x^2+3x+1\right)\left(x^2+2x+2\right)^2-\left(x^2+3x+1\right)^2\frac{d}{dx}\left(\left(x^2+2x+2\right)^2\right)}{\left(x^2+2x+2\right)^{4}}$
6
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{2\left(x^2+3x+1\right)\frac{d}{dx}\left(x^2+3x+1\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\frac{d}{dx}\left(x^2+2x+2\right)}{\left(x^2+2x+2\right)^{4}}$
7
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(1\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\frac{d}{dx}\left(x^2+2x+2\right)}{\left(x^2+2x+2\right)^{4}}$
8
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(1\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(2\right)\right)}{\left(x^2+2x+2\right)^{4}}$
9
The derivative of the constant function ($1$) is equal to zero
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(3x\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(2\right)\right)}{\left(x^2+2x+2\right)^{4}}$
10
The derivative of the constant function ($2$) is equal to zero
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(3x\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)}{\left(x^2+2x+2\right)^{4}}$
Intermediate steps
The derivative of a function multiplied by a constant ($3$) is equal to the constant times the derivative of the function
$3\frac{d}{dx}\left(x\right)$
The derivative of the linear function is equal to $1$
$3$
11
The derivative of the linear function times a constant, is equal to the constant
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+3\frac{d}{dx}\left(x\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)}{\left(x^2+2x+2\right)^{4}}$
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Intermediate steps
The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function
$2\frac{d}{dx}\left(x\right)$
The derivative of the linear function is equal to $1$
$2$
12
The derivative of the linear function times a constant, is equal to the constant
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+3\frac{d}{dx}\left(x\right)\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+2\frac{d}{dx}\left(x\right)\right)}{\left(x^2+2x+2\right)^{4}}$
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Intermediate steps
The derivative of the linear function is equal to $1$
$3\cdot 1$
Any expression multiplied by $1$ is equal to itself
$3$
13
The derivative of the linear function is equal to $1$
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+2\frac{d}{dx}\left(x\right)\right)}{\left(x^2+2x+2\right)^{4}}$
Explain more
Intermediate steps
The derivative of the linear function is equal to $1$
$3\cdot 1$
Any expression multiplied by $1$ is equal to itself
$3$
The derivative of the linear function is equal to $1$
$2\cdot 1$
Any expression multiplied by $1$ is equal to itself
$2$
14
The derivative of the linear function is equal to $1$
$\frac{2\left(x^2+3x+1\right)\left(\frac{d}{dx}\left(x^2\right)+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+2\right)}{\left(x^2+2x+2\right)^{4}}$
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Intermediate steps
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$2x^{\left(2-1\right)}$
Subtract the values $2$ and $-1$
$2x$
15
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{2\left(x^2+3x+1\right)\left(2x+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(\frac{d}{dx}\left(x^2\right)+2\right)}{\left(x^2+2x+2\right)^{4}}$
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Intermediate steps
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$2x^{\left(2-1\right)}$
Subtract the values $2$ and $-1$
$2x$
16
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{2\left(x^2+3x+1\right)\left(2x+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(2x+2\right)}{\left(x^2+2x+2\right)^{4}}$
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Final Answer
$\frac{2\left(x^2+3x+1\right)\left(2x+3\right)\left(x^2+2x+2\right)^2-2\left(x^2+3x+1\right)^2\left(x^2+2x+2\right)\left(2x+2\right)}{\left(x^2+2x+2\right)^{4}}$