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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: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

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

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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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 $2$

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

Multiply $2$ times $2$

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

Multiply $2$ times $2$

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

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

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

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

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

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

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

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 $1$

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

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 $1$

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

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 $f'(x) = nx^{n-1}$

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

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 $f'(x) = nx^{n-1}$

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

Final Answer

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

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