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Product rule of differentiation Calculator

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1

Solved example of product rule of differentiation

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=3x+2$ and $g=x^2-1$

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

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

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

$\left(\frac{d}{dx}\left(3x\right)+0\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2-1\right)$

$x+0=x$, where $x$ is any expression

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

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

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

The derivative of a function multiplied by a constant ($3$) is equal to the constant times the derivative of the function

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

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

$3\left(x^2-1\right)$
5

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

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

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

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

$\left(\frac{d}{dx}\left(3x\right)+0\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2-1\right)$

$x+0=x$, where $x$ is any expression

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

$3\left(x^2-1\right)+\left(3x+2\right)\left(\frac{d}{dx}\left(x^2\right)+0\right)$

$x+0=x$, where $x$ is any expression

$3\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2\right)$
7

The derivative of the constant function ($-1$) is equal to zero

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

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

Subtract the values $2$ and $-1$

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

Any expression to the power of $1$ is equal to that same expression

$3\left(x^2-1\right)+2x\left(3x+2\right)$
8

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

$3\left(x^2-1\right)+2x\left(3x+2\right)$

Final Answer

$3\left(x^2-1\right)+2x\left(3x+2\right)$

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