Calculus Examples

$f (x) = x (200 - (\frac{x}{30})) - (61000 + 80 x)$

Step 1

By the Sum Rule, the derivative of $x (200 - \frac{x}{30}) - (61000 + 80 x)$ with respect to $x$ is $\frac{d}{d x} [x (200 - \frac{x}{30})] + \frac{d}{d x} [- (61000 + 80 x)]$ .

$\frac{d}{d x} [x (200 - \frac{x}{30})] + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2

Evaluate $\frac{d}{d x} [x (200 - \frac{x}{30})]$ .

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Step 2.1

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 $f (x) = x$ and $g (x) = 200 - \frac{x}{30}$ .

$x \frac{d}{d x} [200 - \frac{x}{30}] + (200 - \frac{x}{30}) \frac{d}{d x} [x] + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.2

By the Sum Rule, the derivative of $200 - \frac{x}{30}$ with respect to $x$ is $\frac{d}{d x} [200] + \frac{d}{d x} [- \frac{x}{30}]$ .

$x (\frac{d}{d x} [200] + \frac{d}{d x} [- \frac{x}{30}]) + (200 - \frac{x}{30}) \frac{d}{d x} [x] + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.3

Since $200$ is constant with respect to $x$ , the derivative of $200$ with respect to $x$ is $0$ .

$x (0 + \frac{d}{d x} [- \frac{x}{30}]) + (200 - \frac{x}{30}) \frac{d}{d x} [x] + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.4

Since $- \frac{1}{30}$ is constant with respect to $x$ , the derivative of $- \frac{x}{30}$ with respect to $x$ is $- \frac{1}{30} \frac{d}{d x} [x]$ .

$x (0 - \frac{1}{30} \frac{d}{d x} [x]) + (200 - \frac{x}{30}) \frac{d}{d x} [x] + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.5

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

$x (0 - \frac{1}{30} \cdot 1) + (200 - \frac{x}{30}) \frac{d}{d x} [x] + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.6

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

$x (0 - \frac{1}{30} \cdot 1) + (200 - \frac{x}{30}) \cdot 1 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.7

Multiply $- 1$ by $1$ .

$x (0 - \frac{1}{30}) + (200 - \frac{x}{30}) \cdot 1 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.8

Subtract $\frac{1}{30}$ from $0$ .

$x (- \frac{1}{30}) + (200 - \frac{x}{30}) \cdot 1 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.9

Combine $x$ and $\frac{1}{30}$ .

$- \frac{x}{30} + (200 - \frac{x}{30}) \cdot 1 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.10

Multiply $200 - \frac{x}{30}$ by $1$ .

$- \frac{x}{30} + 200 - \frac{x}{30} + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.11

Subtract $\frac{x}{30}$ from $- \frac{x}{30}$ .

$- 2 \frac{x}{30} + 200 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.12

Combine $- 2$ and $\frac{x}{30}$ .

$\frac{- 2 x}{30} + 200 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.13

Cancel the common factor of $- 2$ and $30$ .

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Step 2.13.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x)}{30} + 200 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.13.2

Cancel the common factors.

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Step 2.13.2.1

Factor $2$ out of $30$ .

$\frac{2 (- x)}{2 (15)} + 200 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.13.2.2

Cancel the common factor.

$\frac{2 (- x)}{2 \cdot 15} + 200 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.13.2.3

Rewrite the expression.

$\frac{- x}{15} + 200 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 2.14

Move the negative in front of the fraction.

$- \frac{x}{15} + 200 + \frac{d}{d x} [- (61000 + 80 x)]$

Step 3

Evaluate $\frac{d}{d x} [- (61000 + 80 x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (61000 + 80 x)$ with respect to $x$ is $- \frac{d}{d x} [61000 + 80 x]$ .

$- \frac{x}{15} + 200 - \frac{d}{d x} [61000 + 80 x]$

Step 3.2

By the Sum Rule, the derivative of $61000 + 80 x$ with respect to $x$ is $\frac{d}{d x} [61000] + \frac{d}{d x} [80 x]$ .

$- \frac{x}{15} + 200 - (\frac{d}{d x} [61000] + \frac{d}{d x} [80 x])$

Step 3.3

Since $61000$ is constant with respect to $x$ , the derivative of $61000$ with respect to $x$ is $0$ .

$- \frac{x}{15} + 200 - (0 + \frac{d}{d x} [80 x])$

Step 3.4

Since $80$ is constant with respect to $x$ , the derivative of $80 x$ with respect to $x$ is $80 \frac{d}{d x} [x]$ .

$- \frac{x}{15} + 200 - (0 + 80 \frac{d}{d x} [x])$

Step 3.5

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

$- \frac{x}{15} + 200 - (0 + 80 \cdot 1)$

Step 3.6

Multiply $80$ by $1$ .

$- \frac{x}{15} + 200 - (0 + 80)$

Step 3.7

Add $0$ and $80$ .

$- \frac{x}{15} + 200 - 1 \cdot 80$

Step 3.8

Multiply $- 1$ by $80$ .

$- \frac{x}{15} + 200 - 80$

Step 4

Subtract $80$ from $200$ .

$- \frac{x}{15} + 120$