Calculus Examples

$f (x) = \frac{- 3 x^{5} + 40 x^{3} + 135 x}{270}$

Step 1

Since $\frac{1}{270}$ is constant with respect to $x$ , the derivative of $\frac{- 3 x^{5} + 40 x^{3} + 135 x}{270}$ with respect to $x$ is $\frac{1}{270} \frac{d}{d x} [- 3 x^{5} + 40 x^{3} + 135 x]$ .

$\frac{1}{270} \frac{d}{d x} [- 3 x^{5} + 40 x^{3} + 135 x]$

Step 2

By the Sum Rule, the derivative of $- 3 x^{5} + 40 x^{3} + 135 x$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [135 x]$ .

$\frac{1}{270} (\frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [135 x])$

Step 3

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

$\frac{1}{270} (- 3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [135 x])$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{270} (- 3 (5 x^{4}) + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [135 x])$

Step 5

Multiply $5$ by $- 3$ .

$\frac{1}{270} (- 15 x^{4} + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [135 x])$

Step 6

Since $40$ is constant with respect to $x$ , the derivative of $40 x^{3}$ with respect to $x$ is $40 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{270} (- 15 x^{4} + 40 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [135 x])$

Step 7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{270} (- 15 x^{4} + 40 (3 x^{2}) + \frac{d}{d x} [135 x])$

Step 8

Multiply $3$ by $40$ .

$\frac{1}{270} (- 15 x^{4} + 120 x^{2} + \frac{d}{d x} [135 x])$

Step 9

Since $135$ is constant with respect to $x$ , the derivative of $135 x$ with respect to $x$ is $135 \frac{d}{d x} [x]$ .

$\frac{1}{270} (- 15 x^{4} + 120 x^{2} + 135 \frac{d}{d x} [x])$

Step 10

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

$\frac{1}{270} (- 15 x^{4} + 120 x^{2} + 135 \cdot 1)$

Step 11

Multiply $135$ by $1$ .

$\frac{1}{270} (- 15 x^{4} + 120 x^{2} + 135)$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$\frac{1}{270} (- 15 x^{4}) + \frac{1}{270} (120 x^{2}) + \frac{1}{270} \cdot 135$

Step 12.2

Combine terms.

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Step 12.2.1

Combine $- 15$ and $\frac{1}{270}$ .

$\frac{- 15}{270} x^{4} + \frac{1}{270} (120 x^{2}) + \frac{1}{270} \cdot 135$

Step 12.2.2

Combine $\frac{- 15}{270}$ and $x^{4}$ .

$\frac{- 15 x^{4}}{270} + \frac{1}{270} (120 x^{2}) + \frac{1}{270} \cdot 135$

Step 12.2.3

Cancel the common factor of $- 15$ and $270$ .

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Step 12.2.3.1

Factor $15$ out of $- 15 x^{4}$ .

$\frac{15 (- x^{4})}{270} + \frac{1}{270} (120 x^{2}) + \frac{1}{270} \cdot 135$

Step 12.2.3.2

Cancel the common factors.

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Step 12.2.3.2.1

Factor $15$ out of $270$ .

$\frac{15 (- x^{4})}{15 (18)} + \frac{1}{270} (120 x^{2}) + \frac{1}{270} \cdot 135$

Step 12.2.3.2.2

Cancel the common factor.

$\frac{15 (- x^{4})}{15 \cdot 18} + \frac{1}{270} (120 x^{2}) + \frac{1}{270} \cdot 135$

Step 12.2.3.2.3

Rewrite the expression.

$\frac{- x^{4}}{18} + \frac{1}{270} (120 x^{2}) + \frac{1}{270} \cdot 135$

Step 12.2.4

Move the negative in front of the fraction.

$- \frac{x^{4}}{18} + \frac{1}{270} (120 x^{2}) + \frac{1}{270} \cdot 135$

Step 12.2.5

Combine $120$ and $\frac{1}{270}$ .

$- \frac{x^{4}}{18} + \frac{120}{270} x^{2} + \frac{1}{270} \cdot 135$

Step 12.2.6

Combine $\frac{120}{270}$ and $x^{2}$ .

$- \frac{x^{4}}{18} + \frac{120 x^{2}}{270} + \frac{1}{270} \cdot 135$

Step 12.2.7

Cancel the common factor of $120$ and $270$ .

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Step 12.2.7.1

Factor $30$ out of $120 x^{2}$ .

$- \frac{x^{4}}{18} + \frac{30 (4 x^{2})}{270} + \frac{1}{270} \cdot 135$

Step 12.2.7.2

Cancel the common factors.

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Step 12.2.7.2.1

Factor $30$ out of $270$ .

$- \frac{x^{4}}{18} + \frac{30 (4 x^{2})}{30 (9)} + \frac{1}{270} \cdot 135$

Step 12.2.7.2.2

Cancel the common factor.

$- \frac{x^{4}}{18} + \frac{30 (4 x^{2})}{30 \cdot 9} + \frac{1}{270} \cdot 135$

Step 12.2.7.2.3

Rewrite the expression.

$- \frac{x^{4}}{18} + \frac{4 x^{2}}{9} + \frac{1}{270} \cdot 135$

Step 12.2.8

Combine $\frac{1}{270}$ and $135$ .

$- \frac{x^{4}}{18} + \frac{4 x^{2}}{9} + \frac{135}{270}$

Step 12.2.9

Cancel the common factor of $135$ and $270$ .

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Step 12.2.9.1

Factor $135$ out of $135$ .

$- \frac{x^{4}}{18} + \frac{4 x^{2}}{9} + \frac{135 (1)}{270}$

Step 12.2.9.2

Cancel the common factors.

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Step 12.2.9.2.1

Factor $135$ out of $270$ .

$- \frac{x^{4}}{18} + \frac{4 x^{2}}{9} + \frac{135 \cdot 1}{135 \cdot 2}$

Step 12.2.9.2.2

Cancel the common factor.

$- \frac{x^{4}}{18} + \frac{4 x^{2}}{9} + \frac{135 \cdot 1}{135 \cdot 2}$

Step 12.2.9.2.3

Rewrite the expression.

$- \frac{x^{4}}{18} + \frac{4 x^{2}}{9} + \frac{1}{2}$