Final answer to the problem
$19+12x$
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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 product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=
$\frac{d}{dx}\left(3x+5\right)\left(2x+3\right)+\left(3x+5\right)\frac{d}{dx}\left(2x+3\right)$
2
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\left(\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(5\right)\right)\left(2x+3\right)+\left(3x+5\right)\frac{d}{dx}\left(2x+3\right)$
3
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\left(\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(5\right)\right)\left(2x+3\right)+\left(3x+5\right)\left(\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(3\right)\right)$
4
The derivative of the constant function ($5$) is equal to zero
$\frac{d}{dx}\left(3x\right)\left(2x+3\right)+\left(3x+5\right)\left(\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(3\right)\right)$
5
The derivative of the constant function ($3$) is equal to zero
$\frac{d}{dx}\left(3x\right)\left(2x+3\right)+\left(3x+5\right)\frac{d}{dx}\left(2x\right)$
Intermediate steps
6
The derivative of the linear function times a constant, is equal to the constant
$3\frac{d}{dx}\left(x\right)\left(2x+3\right)+\left(3x+5\right)\frac{d}{dx}\left(2x\right)$
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Intermediate steps
7
The derivative of the linear function times a constant, is equal to the constant
$3\frac{d}{dx}\left(x\right)\left(2x+3\right)+2\left(3x+5\right)\frac{d}{dx}\left(x\right)$
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Intermediate steps
8
The derivative of the linear function is equal to $1$
$3\left(2x+3\right)+2\left(3x+5\right)\frac{d}{dx}\left(x\right)$
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Intermediate steps
9
The derivative of the linear function is equal to $1$
$3\left(2x+3\right)+2\left(3x+5\right)$
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Intermediate steps
10
Simplify the derivative
$19+12x$
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Final answer to the problem
$19+12x$