1
Here, we show you a step-by-step solved example of product rule of differentiation. This solution was automatically generated by our smart calculator:
$\frac{d}{dx}\left(\left(3x+2\right)\left(x^2-1\right)\right)$
2
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=
$\frac{d}{dx}\left(3x+2\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2-1\right)$
3
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\left(\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(2\right)\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2-1\right)$
4
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\left(\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(2\right)\right)\left(x^2-1\right)+\left(3x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-1\right)\right)$
5
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=
$\left(\frac{d}{dx}\left(3\right)x+3\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(2\right)\right)\left(x^2-1\right)+\left(3x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-1\right)\right)$
6
The derivative of the constant function ($3$) is equal to zero
$\left(0x+3\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(2\right)\right)\left(x^2-1\right)+\left(3x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-1\right)\right)$
7
The derivative of the constant function ($2$) is equal to zero
$\left(0x+3\frac{d}{dx}\left(x\right)\right)\left(x^2-1\right)+\left(3x+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-1\right)\right)$
8
The derivative of the constant function ($-1$) is equal to zero
$\left(0x+3\frac{d}{dx}\left(x\right)\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2\right)$
9
Any expression multiplied by $0$ is equal to $0$
$\left(0+3\frac{d}{dx}\left(x\right)\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2\right)$
10
$x+0=x$, where $x$ is any expression
$3\frac{d}{dx}\left(x\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2\right)$
Intermediate steps
The derivative of the linear function is equal to $1$
$3\cdot 1\left(x^2-1\right)$
Any expression multiplied by $1$ is equal to itself
$3\left(x^2-1\right)$
11
The derivative of the linear function is equal to $1$
$3\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2\right)$
Explain this step further
Intermediate steps
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$2\left(3x+2\right)x^{\left(2-1\right)}$
Subtract the values $2$ and $-1$
$2\left(3x+2\right)x$
12
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$3\left(x^2-1\right)+2\left(3x+2\right)x$
Explain this step further
Intermediate steps
Multiply the single term $3$ by each term of the polynomial $\left(x^2-1\right)$
$3x^2-1\cdot 3+2\left(3x+2\right)x$
$3x^2-3+2\left(3x+2\right)x$
Solve the product $2\left(3x+2\right)x$
$3x^2-3+\left(6x+4\right)x$
Multiply the single term $x$ by each term of the polynomial $\left(6x+4\right)$
$3x^2-3+6x\cdot x+4x$
When multiplying two powers that have the same base ($x$), you can add the exponents
$3x^2-3+6x^2+4x$
Combining like terms $3x^2$ and $6x^2$
$9x^2-3+4x$
13
Simplify the derivative
$9x^2-3+4x$
Explain this step further
Final answer to the problem
$9x^2-3+4x$