Calculus Examples

$600 - 0.5 x - (15000 + 130 x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $600 - 0.5 x - (15000 + 130 x)$ with respect to $x$ is $\frac{d}{d x} [600] + \frac{d}{d x} [- 0.5 x] + \frac{d}{d x} [- (15000 + 130 x)]$ .

$\frac{d}{d x} [600] + \frac{d}{d x} [- 0.5 x] + \frac{d}{d x} [- (15000 + 130 x)]$

Step 1.2

Since $600$ is constant with respect to $x$ , the derivative of $600$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 0.5 x] + \frac{d}{d x} [- (15000 + 130 x)]$

Step 2

Evaluate $\frac{d}{d x} [- 0.5 x]$ .

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Step 2.1

Since $- 0.5$ is constant with respect to $x$ , the derivative of $- 0.5 x$ with respect to $x$ is $- 0.5 \frac{d}{d x} [x]$ .

$0 - 0.5 \frac{d}{d x} [x] + \frac{d}{d x} [- (15000 + 130 x)]$

Step 2.2

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

$0 - 0.5 \cdot 1 + \frac{d}{d x} [- (15000 + 130 x)]$

Step 2.3

Multiply $- 0.5$ by $1$ .

$0 - 0.5 + \frac{d}{d x} [- (15000 + 130 x)]$

Step 3

Evaluate $\frac{d}{d x} [- (15000 + 130 x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (15000 + 130 x)$ with respect to $x$ is $- \frac{d}{d x} [15000 + 130 x]$ .

$0 - 0.5 - \frac{d}{d x} [15000 + 130 x]$

Step 3.2

By the Sum Rule, the derivative of $15000 + 130 x$ with respect to $x$ is $\frac{d}{d x} [15000] + \frac{d}{d x} [130 x]$ .

$0 - 0.5 - (\frac{d}{d x} [15000] + \frac{d}{d x} [130 x])$

Step 3.3

Since $15000$ is constant with respect to $x$ , the derivative of $15000$ with respect to $x$ is $0$ .

$0 - 0.5 - (0 + \frac{d}{d x} [130 x])$

Step 3.4

Since $130$ is constant with respect to $x$ , the derivative of $130 x$ with respect to $x$ is $130 \frac{d}{d x} [x]$ .

$0 - 0.5 - (0 + 130 \frac{d}{d x} [x])$

Step 3.5

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

$0 - 0.5 - (0 + 130 \cdot 1)$

Step 3.6

Multiply $130$ by $1$ .

$0 - 0.5 - (0 + 130)$

Step 3.7

Add $0$ and $130$ .

$0 - 0.5 - 1 \cdot 130$

Step 3.8

Multiply $- 1$ by $130$ .

$0 - 0.5 - 130$

Step 4

Combine terms.

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Step 4.1

Subtract $0.5$ from $0$ .

$- 0.5 - 130$

Step 4.2

Subtract $130$ from $- 0.5$ .

$- 130.5$