Calculus Examples

$- (x^{2} + 8 x + 12)$

Step 1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x^{2} + 8 x + 12)$ with respect to $x$ is $- \frac{d}{d x} [x^{2} + 8 x + 12]$ .

$- \frac{d}{d x} [x^{2} + 8 x + 12]$

Step 2

By the Sum Rule, the derivative of $x^{2} + 8 x + 12$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [12]$ .

$- (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [12])$

Step 3

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

$- (2 x + \frac{d}{d x} [8 x] + \frac{d}{d x} [12])$

Step 4

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$- (2 x + 8 \frac{d}{d x} [x] + \frac{d}{d x} [12])$

Step 5

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

$- (2 x + 8 \cdot 1 + \frac{d}{d x} [12])$

Step 6

Multiply $8$ by $1$ .

$- (2 x + 8 + \frac{d}{d x} [12])$

Step 7

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

$- (2 x + 8 + 0)$

Step 8

Add $2 x + 8$ and $0$ .

$- (2 x + 8)$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$- (2 x) - 1 \cdot 8$

Step 9.2

Combine terms.

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Step 9.2.1

Multiply $2$ by $- 1$ .

$- 2 x - 1 \cdot 8$

Step 9.2.2

Multiply $- 1$ by $8$ .

$- 2 x - 8$