Calculus Examples

$P (x) = (8 x - 0.06 x^{2}) - (186 + 6.9 x)$

Step 1

By the Sum Rule, the derivative of $(8 x - 0.06 x^{2}) - (186 + 6.9 x)$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [- 0.06 x^{2}] + \frac{d}{d x} [- (186 + 6.9 x)]$ .

$\frac{d}{d x} [8 x] + \frac{d}{d x} [- 0.06 x^{2}] + \frac{d}{d x} [- (186 + 6.9 x)]$

Step 2

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 2.1

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

$8 \frac{d}{d x} [x] + \frac{d}{d x} [- 0.06 x^{2}] + \frac{d}{d x} [- (186 + 6.9 x)]$

Step 2.2

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

$8 \cdot 1 + \frac{d}{d x} [- 0.06 x^{2}] + \frac{d}{d x} [- (186 + 6.9 x)]$

Step 2.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d x} [- 0.06 x^{2}] + \frac{d}{d x} [- (186 + 6.9 x)]$

Step 3

Evaluate $\frac{d}{d x} [- 0.06 x^{2}]$ .

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Step 3.1

Since $- 0.06$ is constant with respect to $x$ , the derivative of $- 0.06 x^{2}$ with respect to $x$ is $- 0.06 \frac{d}{d x} [x^{2}]$ .

$8 - 0.06 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- (186 + 6.9 x)]$

Step 3.2

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

$8 - 0.06 (2 x) + \frac{d}{d x} [- (186 + 6.9 x)]$

Step 3.3

Multiply $2$ by $- 0.06$ .

$8 - 0.12 x + \frac{d}{d x} [- (186 + 6.9 x)]$

Step 4

Evaluate $\frac{d}{d x} [- (186 + 6.9 x)]$ .

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Step 4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (186 + 6.9 x)$ with respect to $x$ is $- \frac{d}{d x} [186 + 6.9 x]$ .

$8 - 0.12 x - \frac{d}{d x} [186 + 6.9 x]$

Step 4.2

By the Sum Rule, the derivative of $186 + 6.9 x$ with respect to $x$ is $\frac{d}{d x} [186] + \frac{d}{d x} [6.9 x]$ .

$8 - 0.12 x - (\frac{d}{d x} [186] + \frac{d}{d x} [6.9 x])$

Step 4.3

Since $186$ is constant with respect to $x$ , the derivative of $186$ with respect to $x$ is $0$ .

$8 - 0.12 x - (0 + \frac{d}{d x} [6.9 x])$

Step 4.4

Since $6.9$ is constant with respect to $x$ , the derivative of $6.9 x$ with respect to $x$ is $6.9 \frac{d}{d x} [x]$ .

$8 - 0.12 x - (0 + 6.9 \frac{d}{d x} [x])$

Step 4.5

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

$8 - 0.12 x - (0 + 6.9 \cdot 1)$

Step 4.6

Multiply $6.9$ by $1$ .

$8 - 0.12 x - (0 + 6.9)$

Step 4.7

Add $0$ and $6.9$ .

$8 - 0.12 x - 1 \cdot 6.9$

Step 4.8

Multiply $- 1$ by $6.9$ .

$8 - 0.12 x - 6.9$

Step 5

Subtract $6.9$ from $8$ .

$- 0.12 x + 1.1$