Calculus Examples

$f (x) = 300 - x - \frac{3 x}{x + 2}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $300 - x - \frac{3 x}{x + 2}$ with respect to $x$ is $\frac{d}{d x} [300] + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{3 x}{x + 2}]$ .

$\frac{d}{d x} [300] + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{3 x}{x + 2}]$

Step 1.2

Since $300$ is constant with respect to $x$ , the derivative of $300$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{3 x}{x + 2}]$

Step 2

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

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Step 2.1

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

$0 - \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{3 x}{x + 2}]$

Step 2.2

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

$0 - 1 \cdot 1 + \frac{d}{d x} [- \frac{3 x}{x + 2}]$

Step 2.3

Multiply $- 1$ by $1$ .

$0 - 1 + \frac{d}{d x} [- \frac{3 x}{x + 2}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{3 x}{x + 2}]$ .

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3 x}{x + 2}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{x}{x + 2}]$ .

$0 - 1 - 3 \frac{d}{d x} [\frac{x}{x + 2}]$

Step 3.2

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

$0 - 1 - 3 \frac{(x + 2) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.3

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

$0 - 1 - 3 \frac{(x + 2) \cdot 1 - x \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.4

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

$0 - 1 - 3 \frac{(x + 2) \cdot 1 - x (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 3.5

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

$0 - 1 - 3 \frac{(x + 2) \cdot 1 - x (1 + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 3.6

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

$0 - 1 - 3 \frac{(x + 2) \cdot 1 - x (1 + 0)}{{(x + 2)}^{2}}$

Step 3.7

Multiply $x + 2$ by $1$ .

$0 - 1 - 3 \frac{x + 2 - x (1 + 0)}{{(x + 2)}^{2}}$

Step 3.8

Add $1$ and $0$ .

$0 - 1 - 3 \frac{x + 2 - x \cdot 1}{{(x + 2)}^{2}}$

Step 3.9

Multiply $- 1$ by $1$ .

$0 - 1 - 3 \frac{x + 2 - x}{{(x + 2)}^{2}}$

Step 3.10

Subtract $x$ from $x$ .

$0 - 1 - 3 \frac{0 + 2}{{(x + 2)}^{2}}$

Step 3.11

Add $0$ and $2$ .

$0 - 1 - 3 \frac{2}{{(x + 2)}^{2}}$

Step 3.12

Combine $- 3$ and $\frac{2}{{(x + 2)}^{2}}$ .

$0 - 1 + \frac{- 3 \cdot 2}{{(x + 2)}^{2}}$

Step 3.13

Multiply $- 3$ by $2$ .

$0 - 1 + \frac{- 6}{{(x + 2)}^{2}}$

Step 3.14

Move the negative in front of the fraction.

$0 - 1 - \frac{6}{{(x + 2)}^{2}}$

Step 4

Simplify.

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Step 4.1

Subtract $1$ from $0$ .

$- 1 - \frac{6}{{(x + 2)}^{2}}$

Step 4.2

Reorder terms.

$- \frac{6}{{(x + 2)}^{2}} - 1$