Calculus Examples

$p (x) = 4000 (15 - x) (x - 2)$

Step 1

Since $4000$ is constant with respect to $x$ , the derivative of $4000 (15 - x) (x - 2)$ with respect to $x$ is $4000 \frac{d}{d x} [(15 - x) (x - 2)]$ .

$4000 \frac{d}{d x} [(15 - x) (x - 2)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 15 - x$ and $g (x) = x - 2$ .

$4000 ((15 - x) \frac{d}{d x} [x - 2] + (x - 2) \frac{d}{d x} [15 - x])$

Step 3

Differentiate.

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Step 3.1

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

$4000 ((15 - x) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [15 - x])$

Step 3.2

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

$4000 ((15 - x) (1 + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [15 - x])$

Step 3.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$4000 ((15 - x) (1 + 0) + (x - 2) \frac{d}{d x} [15 - x])$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$4000 ((15 - x) \cdot 1 + (x - 2) \frac{d}{d x} [15 - x])$

Step 3.4.2

Multiply $15 - x$ by $1$ .

$4000 (15 - x + (x - 2) \frac{d}{d x} [15 - x])$

Step 3.5

By the Sum Rule, the derivative of $15 - x$ with respect to $x$ is $\frac{d}{d x} [15] + \frac{d}{d x} [- x]$ .

$4000 (15 - x + (x - 2) (\frac{d}{d x} [15] + \frac{d}{d x} [- x]))$

Step 3.6

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$4000 (15 - x + (x - 2) (0 + \frac{d}{d x} [- x]))$

Step 3.7

Add $0$ and $\frac{d}{d x} [- x]$ .

$4000 (15 - x + (x - 2) \frac{d}{d x} [- x])$

Step 3.8

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

$4000 (15 - x + (x - 2) (- \frac{d}{d x} [x]))$

Step 3.9

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

$4000 (15 - x + (x - 2) (- 1 \cdot 1))$

Step 3.10

Simplify the expression.

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Step 3.10.1

Multiply $- 1$ by $1$ .

$4000 (15 - x + (x - 2) \cdot - 1)$

Step 3.10.2

Move $- 1$ to the left of $x - 2$ .

$4000 (15 - x - 1 \cdot (x - 2))$

Step 3.10.3

Rewrite $- 1 (x - 2)$ as $- (x - 2)$ .

$4000 (15 - x - (x - 2))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$4000 (15 - x - x - - 2)$

Step 4.2

Apply the distributive property.

$4000 \cdot 15 + 4000 (- x) + 4000 (- x) + 4000 (- - 2)$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $4000$ by $15$ .

$60000 + 4000 (- x) + 4000 (- x) + 4000 (- - 2)$

Step 4.3.2

Multiply $- 1$ by $4000$ .

$60000 - 4000 x + 4000 (- x) + 4000 (- - 2)$

Step 4.3.3

Multiply $- 1$ by $4000$ .

$60000 - 4000 x - 4000 x + 4000 (- - 2)$

Step 4.3.4

Multiply $- 1$ by $- 2$ .

$60000 - 4000 x - 4000 x + 4000 \cdot 2$

Step 4.3.5

Multiply $4000$ by $2$ .

$60000 - 4000 x - 4000 x + 8000$

Step 4.3.6

Subtract $4000 x$ from $- 4000 x$ .

$60000 - 8000 x + 8000$

Step 4.3.7

Add $60000$ and $8000$ .

$- 8000 x + 68000$