Find the Derivative Using Quotient Rule - d/dx (x^2+13x+54)/(x+6)

$\frac{x^{2} + 13 x + 54}{x + 6}$

Step 1

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 $f (x) = x^{2} + 13 x + 54$ and $g (x) = x + 6$ .

$\frac{(x + 6) \frac{d}{d x} [x^{2} + 13 x + 54] - (x^{2} + 13 x + 54) \frac{d}{d x} [x + 6]}{{(x + 6)}^{2}}$

Step 2

Differentiate.

By the Sum Rule, the derivative of $x^{2} + 13 x + 54$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [13 x] + \frac{d}{d x} [54]$ .

$\frac{(x + 6) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [13 x] + \frac{d}{d x} [54]) - (x^{2} + 13 x + 54) \frac{d}{d x} [x + 6]}{{(x + 6)}^{2}}$

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

$\frac{(x + 6) (2 x + \frac{d}{d x} [13 x] + \frac{d}{d x} [54]) - (x^{2} + 13 x + 54) \frac{d}{d x} [x + 6]}{{(x + 6)}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [13 x]$ .

Since $13$ is constant with respect to $x$ , the derivative of $13 x$ with respect to $x$ is $13 \frac{d}{d x} [x]$ .

$\frac{(x + 6) (2 x + 13 \frac{d}{d x} [x] + \frac{d}{d x} [54]) - (x^{2} + 13 x + 54) \frac{d}{d x} [x + 6]}{{(x + 6)}^{2}}$

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

$\frac{(x + 6) (2 x + 13 \cdot 1 + \frac{d}{d x} [54]) - (x^{2} + 13 x + 54) \frac{d}{d x} [x + 6]}{{(x + 6)}^{2}}$

Multiply $13$ by $1$ .

$\frac{(x + 6) (2 x + 13 + \frac{d}{d x} [54]) - (x^{2} + 13 x + 54) \frac{d}{d x} [x + 6]}{{(x + 6)}^{2}}$

Step 4

Differentiate.

Since $54$ is constant with respect to $x$ , the derivative of $54$ with respect to $x$ is $0$ .

$\frac{(x + 6) (2 x + 13 + 0) - (x^{2} + 13 x + 54) \frac{d}{d x} [x + 6]}{{(x + 6)}^{2}}$

Add $2 x + 13$ and $0$ .

$\frac{(x + 6) (2 x + 13) - (x^{2} + 13 x + 54) \frac{d}{d x} [x + 6]}{{(x + 6)}^{2}}$

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$\frac{(x + 6) (2 x + 13) - (x^{2} + 13 x + 54) (\frac{d}{d x} [x] + \frac{d}{d x} [6])}{{(x + 6)}^{2}}$

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

$\frac{(x + 6) (2 x + 13) - (x^{2} + 13 x + 54) (1 + \frac{d}{d x} [6])}{{(x + 6)}^{2}}$

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{(x + 6) (2 x + 13) - (x^{2} + 13 x + 54) (1 + 0)}{{(x + 6)}^{2}}$

Add $1$ and $0$ .

$\frac{(x + 6) (2 x + 13) - (x^{2} + 13 x + 54) \cdot 1}{{(x + 6)}^{2}}$

Step 5

Simplify.