Calculus Examples

$\frac{2 x (150 - 25 x)}{30}$

Step 1

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

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Step 1.1

Factor $2$ out of $2 x (150 - 25 x)$ .

$\frac{d}{d x} [\frac{2 (x (150 - 25 x))}{30}]$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $30$ .

$\frac{d}{d x} [\frac{2 (x (150 - 25 x))}{2 \cdot 15}]$

Step 1.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{2 (x (150 - 25 x))}{2 \cdot 15}]$

Step 1.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{x (150 - 25 x)}{15}]$

Step 2

Cancel the common factor of $150 - 25 x$ and $15$ .

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Step 2.1

Factor $5$ out of $x (150 - 25 x)$ .

$\frac{d}{d x} [\frac{5 (x (30 - 5 x))}{15}]$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $5$ out of $15$ .

$\frac{d}{d x} [\frac{5 (x (30 - 5 x))}{5 (3)}]$

Step 2.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{5 (x (30 - 5 x))}{5 \cdot 3}]$

Step 2.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{x (30 - 5 x)}{3}]$

Step 3

Factor $5$ out of $30 - 5 x$ .

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Step 3.1

Factor $5$ out of $30$ .

$\frac{d}{d x} [\frac{x (5 (6) - 5 x)}{3}]$

Step 3.2

Factor $5$ out of $- 5 x$ .

$\frac{d}{d x} [\frac{x (5 (6) + 5 (- x))}{3}]$

Step 3.3

Factor $5$ out of $5 (6) + 5 (- x)$ .

$\frac{d}{d x} [\frac{x (5 (6 - x))}{3}]$

$\frac{d}{d x} [\frac{x \cdot 5 (6 - x)}{3}]$

Step 4

Differentiate using the Constant Multiple Rule.

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Step 4.1

Move $5$ to the left of $x$ .

$\frac{d}{d x} [\frac{5 x (6 - x)}{3}]$

Step 4.2

Since $\frac{5}{3}$ is constant with respect to $x$ , the derivative of $\frac{5 x (6 - x)}{3}$ with respect to $x$ is $\frac{5}{3} \frac{d}{d x} [x (6 - x)]$ .

$\frac{5}{3} \frac{d}{d x} [x (6 - x)]$

Step 5

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 $g (x) = 6 - x$ .

$\frac{5}{3} (x \frac{d}{d x} [6 - x] + (6 - x) \frac{d}{d x} [x])$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $6 - x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- x]$ .

$\frac{5}{3} (x (\frac{d}{d x} [6] + \frac{d}{d x} [- x]) + (6 - x) \frac{d}{d x} [x])$

Step 6.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{5}{3} (x (0 + \frac{d}{d x} [- x]) + (6 - x) \frac{d}{d x} [x])$

Step 6.3

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

$\frac{5}{3} (x \frac{d}{d x} [- x] + (6 - x) \frac{d}{d x} [x])$

Step 6.4

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

$\frac{5}{3} (x (- \frac{d}{d x} [x]) + (6 - x) \frac{d}{d x} [x])$

Step 6.5

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

$\frac{5}{3} (x (- 1 \cdot 1) + (6 - x) \frac{d}{d x} [x])$

Step 6.6

Simplify the expression.

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Step 6.6.1

Multiply $- 1$ by $1$ .

$\frac{5}{3} (x \cdot - 1 + (6 - x) \frac{d}{d x} [x])$

Step 6.6.2

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

$\frac{5}{3} (- 1 \cdot x + (6 - x) \frac{d}{d x} [x])$

Step 6.6.3

Rewrite $- 1 x$ as $- x$ .

$\frac{5}{3} (- x + (6 - x) \frac{d}{d x} [x])$

Step 6.7

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

$\frac{5}{3} (- x + (6 - x) \cdot 1)$

Step 6.8

Simplify by adding terms.

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Step 6.8.1

Multiply $6 - x$ by $1$ .

$\frac{5}{3} (- x + 6 - x)$

Step 6.8.2

Subtract $x$ from $- x$ .

$\frac{5}{3} (- 2 x + 6)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{5}{3} (- 2 x) + \frac{5}{3} \cdot 6$

Step 7.2

Combine terms.

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Step 7.2.1

Combine $- 2$ and $\frac{5}{3}$ .

$\frac{- 2 \cdot 5}{3} x + \frac{5}{3} \cdot 6$

Step 7.2.2

Multiply $- 2$ by $5$ .

$\frac{- 10}{3} x + \frac{5}{3} \cdot 6$

Step 7.2.3

Combine $\frac{- 10}{3}$ and $x$ .

$\frac{- 10 x}{3} + \frac{5}{3} \cdot 6$

Step 7.2.4

Move the negative in front of the fraction.

$- \frac{10 x}{3} + \frac{5}{3} \cdot 6$

Step 7.2.5

Combine $\frac{5}{3}$ and $6$ .

$- \frac{10 x}{3} + \frac{5 \cdot 6}{3}$

Step 7.2.6

Multiply $5$ by $6$ .

$- \frac{10 x}{3} + \frac{30}{3}$

Step 7.2.7

Cancel the common factor of $30$ and $3$ .

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Step 7.2.7.1

Factor $3$ out of $30$ .

$- \frac{10 x}{3} + \frac{3 \cdot 10}{3}$

Step 7.2.7.2

Cancel the common factors.

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Step 7.2.7.2.1

Factor $3$ out of $3$ .

$- \frac{10 x}{3} + \frac{3 \cdot 10}{3 (1)}$

Step 7.2.7.2.2

Cancel the common factor.

$- \frac{10 x}{3} + \frac{3 \cdot 10}{3 \cdot 1}$

Step 7.2.7.2.3

Rewrite the expression.

$- \frac{10 x}{3} + \frac{10}{1}$

Step 7.2.7.2.4

Divide $10$ by $1$ .

$- \frac{10 x}{3} + 10$