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Find the derivative of $\frac{x^2+x-2}{x^2+5x+6}$

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Solution

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Final Answer

$\frac{-2+2x^2+5x}{\left(x+2\right)\left(x+3\right)^2}$
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Step-by-step Solution

Problem to solve:

$\frac{d}{dx}\left(\frac{x^2+x-2}{x^2+5x+6}\right)$

Specify the solving method

1

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

$\frac{\frac{d}{dx}\left(x^2+x-2\right)\left(x^2+5x+6\right)-\left(x^2+x-2\right)\frac{d}{dx}\left(x^2+5x+6\right)}{\left(x^2+5x+6\right)^2}$
2

Simplify the product $-(x^2+x-2)$

$\frac{\frac{d}{dx}\left(x^2+x-2\right)\left(x^2+5x+6\right)+\left(-x^2-\left(x-2\right)\right)\frac{d}{dx}\left(x^2+5x+6\right)}{\left(x^2+5x+6\right)^2}$
3

Simplify the product $-(x-2)$

$\frac{\frac{d}{dx}\left(x^2+x-2\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\frac{d}{dx}\left(x^2+5x+6\right)}{\left(x^2+5x+6\right)^2}$
4

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

$\frac{\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-2\right)\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\frac{d}{dx}\left(x^2+5x+6\right)}{\left(x^2+5x+6\right)^2}$
5

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

$\frac{\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-2\right)\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(5x\right)+\frac{d}{dx}\left(6\right)\right)}{\left(x^2+5x+6\right)^2}$
6

The derivative of the constant function ($-2$) is equal to zero

$\frac{\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x\right)\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(5x\right)+\frac{d}{dx}\left(6\right)\right)}{\left(x^2+5x+6\right)^2}$
7

The derivative of the constant function ($6$) is equal to zero

$\frac{\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x\right)\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(5x\right)\right)}{\left(x^2+5x+6\right)^2}$
8

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

$\frac{\left(\frac{d}{dx}\left(x^2\right)+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(5x\right)\right)}{\left(x^2+5x+6\right)^2}$

The derivative of a function multiplied by a constant ($5$) is equal to the constant times the derivative of the function

$5\frac{d}{dx}\left(x\right)$

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

$5$
9

The derivative of the linear function times a constant, is equal to the constant

$\frac{\left(\frac{d}{dx}\left(x^2\right)+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+5\frac{d}{dx}\left(x\right)\right)}{\left(x^2+5x+6\right)^2}$

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

$5\cdot 1$

Any expression multiplied by $1$ is equal to itself

$5$
10

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

$\frac{\left(\frac{d}{dx}\left(x^2\right)+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+5\right)}{\left(x^2+5x+6\right)^2}$

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

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

Subtract the values $2$ and $-1$

$2x$
11

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

$\frac{\left(2x+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+5\right)}{\left(x^2+5x+6\right)^2}$

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

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

Subtract the values $2$ and $-1$

$2x$
12

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

$\frac{\left(2x+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(2x+5\right)}{\left(x^2+5x+6\right)^2}$

Factor the trinomial $\left(x^2+5x+6\right)$ finding two numbers that multiply to form $6$ and added form $5$

$\begin{matrix}\left(2\right)\left(3\right)=6\\ \left(2\right)+\left(3\right)=5\end{matrix}$

Thus

$\frac{\left(2x+1\right)\left(x+2\right)\left(x+3\right)+\left(-x^2-x+2\right)\left(2x+5\right)}{\left(x^2+5x+6\right)^2}$

Factor the trinomial by $-1$ for an easier handling

$\frac{\left(2x+1\right)\left(x+2\right)\left(x+3\right)-\left(x^2+x-2\right)\left(2x+5\right)}{\left(x^2+5x+6\right)^2}$

Factor the trinomial $-\left(x^2+x-2\right)$ finding two numbers that multiply to form $-2$ and added form $1$

$\begin{matrix}\left(-1\right)\left(2\right)=-2\\ \left(-1\right)+\left(2\right)=1\end{matrix}$

Thus

$\frac{\left(2x+1\right)\left(x+2\right)\left(x+3\right)-\left(2x+5\right)\left(\sqrt{1x^2}+2\right)}{\left(x^2+5x+6\right)^2}$

Factor the trinomial $\left(x^2+5x+6\right)$ finding two numbers that multiply to form $6$ and added form $5$

$\begin{matrix}\left(2\right)\left(3\right)=6\\ \left(2\right)+\left(3\right)=5\end{matrix}$

Thus

$\frac{\left(2x+1\right)\left(x+2\right)\left(x+3\right)-\left(2x+5\right)\left(\sqrt{1x^2}+2\right)}{\left(\left(x+2\right)\left(x+3\right)\right)^2}$

The power of a product is equal to the product of it's factors raised to the same power

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

Simplify the product $-(2x+5)$

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

Factor the polynomial $\left(2x+1\right)\left(x+2\right)\left(x+3\right)+\left(-2x-5\right)\left(x+2\right)$ by it's greatest common factor (GCF): $x+2$

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

Simplify the fraction by $x+2$

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

Expand the expression $\left(2x+1\right)\left(x+3\right)-2x-5$ completely and simplify

$\frac{-2+2x^2+5x}{\left(x+2\right)\left(x+3\right)^2}$
13

Simplify the derivative

$\frac{-2+2x^2+5x}{\left(x+2\right)\left(x+3\right)^2}$

Final Answer

$\frac{-2+2x^2+5x}{\left(x+2\right)\left(x+3\right)^2}$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of (x^2+x)/(x^2+5x) using the product ruleFind derivative of (x^2+x)/(x^2+5x) using the quotient ruleFind derivative of (x^2+x)/(x^2+5x) using logarithmic differentiationFind derivative of (x^2+x)/(x^2+5x) using the definition

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Main topic:

Differential Calculus

Used formulas:

6. See formulas

Related topics:

Differential CalculusCalculusQuotient Rule of DifferentiationSum Rule of DifferentiationConstant Rule for DifferentiationPower Rule for DerivativesBasic Differentiation Rules

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