Calculus Examples

$- \frac{1}{3} (x^{- 3} - x^{6})$

Step 1

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{1}{3} (x^{- 3} - x^{6})$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x^{- 3} - x^{6}]$ .

$- \frac{1}{3} \frac{d}{d x} [x^{- 3} - x^{6}]$

Step 2

By the Sum Rule, the derivative of $x^{- 3} - x^{6}$ with respect to $x$ is $\frac{d}{d x} [x^{- 3}] + \frac{d}{d x} [- x^{6}]$ .

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

Step 3

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

$- \frac{1}{3} (- 3 x^{- 4} + \frac{d}{d x} [- x^{6}])$

Step 4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{6}$ with respect to $x$ is $- \frac{d}{d x} [x^{6}]$ .

$- \frac{1}{3} (- 3 x^{- 4} - \frac{d}{d x} [x^{6}])$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$- \frac{1}{3} (- 3 x^{- 4} - (6 x^{5}))$

Step 6

Multiply $6$ by $- 1$ .

$- \frac{1}{3} (- 3 x^{- 4} - 6 x^{5})$

Step 7

Simplify.

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Step 7.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{3} (- 3 \frac{1}{x^{4}} - 6 x^{5})$

Step 7.2

Apply the distributive property.

$- \frac{1}{3} (- 3 \frac{1}{x^{4}}) - \frac{1}{3} (- 6 x^{5})$

Step 7.3

Combine terms.

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Step 7.3.1

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$- \frac{1}{3} \cdot \frac{- 3}{x^{4}} - \frac{1}{3} (- 6 x^{5})$

Step 7.3.2

Move the negative in front of the fraction.

$- \frac{1}{3} (- \frac{3}{x^{4}}) - \frac{1}{3} (- 6 x^{5})$

Step 7.3.3

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{3}) \frac{3}{x^{4}} - \frac{1}{3} (- 6 x^{5})$

Step 7.3.4

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} \cdot \frac{3}{x^{4}} - \frac{1}{3} (- 6 x^{5})$

Step 7.3.5

Multiply $\frac{1}{3}$ by $\frac{3}{x^{4}}$ .

$\frac{3}{3 x^{4}} - \frac{1}{3} (- 6 x^{5})$

Step 7.3.6

Cancel the common factor of $3$ .

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Step 7.3.6.1

Cancel the common factor.

$\frac{3}{3 x^{4}} - \frac{1}{3} (- 6 x^{5})$

Step 7.3.6.2

Rewrite the expression.

$\frac{1}{x^{4}} - \frac{1}{3} (- 6 x^{5})$

Step 7.3.7

Multiply $- 6$ by $- 1$ .

$\frac{1}{x^{4}} + 6 (\frac{1}{3}) x^{5}$

Step 7.3.8

Combine $6$ and $\frac{1}{3}$ .

$\frac{1}{x^{4}} + \frac{6}{3} x^{5}$

Step 7.3.9

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

$\frac{1}{x^{4}} + \frac{6 x^{5}}{3}$

Step 7.3.10

Cancel the common factor of $6$ and $3$ .

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Step 7.3.10.1

Factor $3$ out of $6 x^{5}$ .

$\frac{1}{x^{4}} + \frac{3 (2 x^{5})}{3}$

Step 7.3.10.2

Cancel the common factors.

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Step 7.3.10.2.1

Factor $3$ out of $3$ .

$\frac{1}{x^{4}} + \frac{3 (2 x^{5})}{3 (1)}$

Step 7.3.10.2.2

Cancel the common factor.

$\frac{1}{x^{4}} + \frac{3 (2 x^{5})}{3 \cdot 1}$

Step 7.3.10.2.3

Rewrite the expression.

$\frac{1}{x^{4}} + \frac{2 x^{5}}{1}$

Step 7.3.10.2.4

Divide $2 x^{5}$ by $1$ .

$\frac{1}{x^{4}} + 2 x^{5}$

Step 7.4

Reorder terms.

$2 x^{5} + \frac{1}{x^{4}}$