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

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Solution

Final Answer

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

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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 $1$

$\frac{\left(\frac{d}{dx}\left(x^2\right)+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(5x\right)\right)}{\left(x^2+5x+6\right)^2}$
9

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

$\frac{\left(\frac{d}{dx}\left(x^2\right)+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+5\frac{d}{dx}\left(x\right)\right)}{\left(x^2+5x+6\right)^2}$
10

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

$\frac{\left(\frac{d}{dx}\left(x^2\right)+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+5\right)}{\left(x^2+5x+6\right)^2}$
11

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

$\frac{\left(2x+1\right)\left(x^2+5x+6\right)+\left(-x^2-x+2\right)\left(\frac{d}{dx}\left(x^2\right)+5\right)}{\left(x^2+5x+6\right)^2}$
12

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

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

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x
y
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.
(◻)
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◻/◻
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÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
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log
log◻
lim
d/dx
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∫◻
|◻|
θ
=
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>=
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sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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