Final answer to the problem
$\frac{\left(8x+12\left(x+y\right)+18y\right)\left(3y+2x\right)+5\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$
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Step-by-step Solution
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Find the derivative Find the derivative using the product rule Find the derivative using the quotient rule Logarithmic Differentiation Find the derivative using the definition Suggest another method or feature
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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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
$\frac{\frac{d}{dx}\left(4x^2+12yx+9y^2\right)\left(3y+2x\right)-\left(4x^2+12yx+9y^2\right)\frac{d}{dx}\left(3y+2x\right)}{\left(3y+2x\right)^2}$
2
Simplify the product $-(4x^2+12yx+9y^2)$
$\frac{\frac{d}{dx}\left(4x^2+12yx+9y^2\right)\left(3y+2x\right)+\left(-4x^2-\left(12yx+9y^2\right)\right)\frac{d}{dx}\left(3y+2x\right)}{\left(3y+2x\right)^2}$
3
Simplify the product $-(12yx+9y^2)$
$\frac{\frac{d}{dx}\left(4x^2+12yx+9y^2\right)\left(3y+2x\right)+\left(-4x^2-12yx-9y^2\right)\frac{d}{dx}\left(3y+2x\right)}{\left(3y+2x\right)^2}$
4
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(12yx\right)+\frac{d}{dx}\left(9y^2\right)\right)\left(3y+2x\right)+\left(-4x^2-12yx-9y^2\right)\frac{d}{dx}\left(3y+2x\right)}{\left(3y+2x\right)^2}$
5
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(12yx\right)+\frac{d}{dx}\left(9y^2\right)\right)\left(3y+2x\right)+\left(-4x^2-12yx-9y^2\right)\left(\frac{d}{dx}\left(3y\right)+\frac{d}{dx}\left(2x\right)\right)}{\left(3y+2x\right)^2}$
Intermediate steps
6
The derivative of the linear function times a constant, is equal to the constant
$\frac{\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(12yx\right)+\frac{d}{dx}\left(9y^2\right)\right)\left(3y+2x\right)+\left(-4x^2-12yx-9y^2\right)\left(3\frac{d}{dx}\left(y\right)+\frac{d}{dx}\left(2x\right)\right)}{\left(3y+2x\right)^2}$
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Intermediate steps
7
The derivative of the linear function times a constant, is equal to the constant
$\frac{\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(12yx\right)+\frac{d}{dx}\left(9y^2\right)\right)\left(3y+2x\right)+\left(-4x^2-12yx-9y^2\right)\left(3\frac{d}{dx}\left(y\right)+2\frac{d}{dx}\left(x\right)\right)}{\left(3y+2x\right)^2}$
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Intermediate steps
8
The derivative of the linear function is equal to $1$
$\frac{\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(12yx\right)+\frac{d}{dx}\left(9y^2\right)\right)\left(3y+2x\right)+\left(-4x^2-12yx-9y^2\right)\left(3+2\frac{d}{dx}\left(x\right)\right)}{\left(3y+2x\right)^2}$
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Intermediate steps
9
The derivative of the linear function is equal to $1$
$\frac{\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(12yx\right)+\frac{d}{dx}\left(9y^2\right)\right)\left(3y+2x\right)+\left(3+2\right)\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$
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10
Add the values $3$ and $2$
$\frac{\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(12yx\right)+\frac{d}{dx}\left(9y^2\right)\right)\left(3y+2x\right)+5\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$
Intermediate steps
11
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
$\frac{\left(4\frac{d}{dx}\left(x^2\right)+12\frac{d}{dx}\left(yx\right)+9\frac{d}{dx}\left(y^2\right)\right)\left(3y+2x\right)+5\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$
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12
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=
$\frac{\left(4\frac{d}{dx}\left(x^2\right)+12\left(\frac{d}{dx}\left(y\right)x+y\frac{d}{dx}\left(x\right)\right)+9\frac{d}{dx}\left(y^2\right)\right)\left(3y+2x\right)+5\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$
Intermediate steps
13
The derivative of the linear function is equal to $1$
$\frac{\left(4\frac{d}{dx}\left(x^2\right)+12\left(x+y\frac{d}{dx}\left(x\right)\right)+9\frac{d}{dx}\left(y^2\right)\right)\left(3y+2x\right)+5\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$
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Intermediate steps
14
The derivative of the linear function is equal to $1$
$\frac{\left(4\frac{d}{dx}\left(x^2\right)+12\left(x+y\right)+9\frac{d}{dx}\left(y^2\right)\right)\left(3y+2x\right)+5\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$
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Intermediate steps
15
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{\left(8x+12\left(x+y\right)+18y\right)\left(3y+2x\right)+5\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$
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Final answer to the problem$\frac{\left(8x+12\left(x+y\right)+18y\right)\left(3y+2x\right)+5\left(-4x^2-12yx-9y^2\right)}{\left(3y+2x\right)^2}$