Calculus Examples

$r (x) = 30 (10^{3}) - 120 \cdot 10^{2} + 500$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Raise $10$ to the power of $3$ .

$\frac{d}{d x} [30 \cdot 1000 - 120 \cdot 10^{2} + 500]$

Step 1.2

Raise $10$ to the power of $2$ .

$\frac{d}{d x} [30 \cdot 1000 - 120 \cdot 100 + 500]$

Step 1.3

By the Sum Rule, the derivative of $30 \cdot 1000 - 120 \cdot 100 + 500$ with respect to $x$ is $\frac{d}{d x} [30 \cdot 1000] + \frac{d}{d x} [- 120 \cdot 100] + \frac{d}{d x} [500]$ .

$\frac{d}{d x} [30 \cdot 1000] + \frac{d}{d x} [- 120 \cdot 100] + \frac{d}{d x} [500]$

Step 2

Evaluate $\frac{d}{d x} [30 \cdot 1000]$ .

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Step 2.1

Multiply $30$ by $1000$ .

$\frac{d}{d x} [30000] + \frac{d}{d x} [- 120 \cdot 100] + \frac{d}{d x} [500]$

Step 2.2

Since $30000$ is constant with respect to $x$ , the derivative of $30000$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 120 \cdot 100] + \frac{d}{d x} [500]$

Step 3

Evaluate $\frac{d}{d x} [- 120 \cdot 100]$ .

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Step 3.1

Multiply $- 120$ by $100$ .

$0 + \frac{d}{d x} [- 12000] + \frac{d}{d x} [500]$

Step 3.2

Since $- 12000$ is constant with respect to $x$ , the derivative of $- 12000$ with respect to $x$ is $0$ .

$0 + 0 + \frac{d}{d x} [500]$

Step 4

Since $500$ is constant with respect to $x$ , the derivative of $500$ with respect to $x$ is $0$ .

$0 + 0 + 0$

Step 5

Combine terms.

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Step 5.1

Add $0$ and $0$ .

$0 + 0$

Step 5.2

Add $0$ and $0$ .

$0$