Find the Derivative Using Quotient Rule - d/dx (12x^2-10x-12)÷(3x+2)

$(12 x^{2} - 10 x - 12) \div (3 x + 2)$

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 12 x^{2} - 10 x - 12$ and $g (x) = 3 x + 2$ .

$\frac{(3 x + 2) \frac{d}{d x} [12 x^{2} - 10 x - 12] - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

By the Sum Rule, the derivative of $12 x^{2} - 10 x - 12$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [- 12]$ .

$\frac{(3 x + 2) (\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [- 12]) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$\frac{(3 x + 2) (12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [- 12]) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{(3 x + 2) (12 (2 x) + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [- 12]) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Multiply $2$ by $12$ .

$\frac{(3 x + 2) (24 x + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [- 12]) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Evaluate $\frac{d}{d x} [- 10 x]$ .

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$\frac{(3 x + 2) (24 x - 10 \frac{d}{d x} [x] + \frac{d}{d x} [- 12]) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(3 x + 2) (24 x - 10 \cdot 1 + \frac{d}{d x} [- 12]) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Multiply $- 10$ by $1$ .

$\frac{(3 x + 2) (24 x - 10 + \frac{d}{d x} [- 12]) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Differentiate.

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$\frac{(3 x + 2) (24 x - 10 + 0) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Add $24 x - 10$ and $0$ .

$\frac{(3 x + 2) (24 x - 10) - (12 x^{2} - 10 x - 12) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

By the Sum Rule, the derivative of $3 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

$\frac{(3 x + 2) (24 x - 10) - (12 x^{2} - 10 x - 12) (\frac{d}{d x} [3 x] + \frac{d}{d x} [2])}{{(3 x + 2)}^{2}}$

Evaluate $\frac{d}{d x} [3 x]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{(3 x + 2) (24 x - 10) - (12 x^{2} - 10 x - 12) (3 \frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(3 x + 2)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(3 x + 2) (24 x - 10) - (12 x^{2} - 10 x - 12) (3 \cdot 1 + \frac{d}{d x} [2])}{{(3 x + 2)}^{2}}$

Multiply $3$ by $1$ .

$\frac{(3 x + 2) (24 x - 10) - (12 x^{2} - 10 x - 12) (3 + \frac{d}{d x} [2])}{{(3 x + 2)}^{2}}$

Differentiate using the Constant Rule.

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(3 x + 2) (24 x - 10) - (12 x^{2} - 10 x - 12) (3 + 0)}{{(3 x + 2)}^{2}}$

Add $3$ and $0$ .

$\frac{(3 x + 2) (24 x - 10) - (12 x^{2} - 10 x - 12) \cdot 3}{{(3 x + 2)}^{2}}$

Simplify.