Calculus Examples

$\frac{1}{3 x^{3}} + \frac{x^{7}}{10}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{3 x^{3}} + \frac{x^{7}}{10}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3 x^{3}}] + \frac{d}{d x} [\frac{x^{7}}{10}]$ .

$\frac{d}{d x} [\frac{1}{3 x^{3}}] + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{3 x^{3}}]$ .

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Step 2.1

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

$\frac{1}{3} \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$\frac{1}{3} \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$\frac{1}{3} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.3.2

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

$\frac{1}{3} (- u^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$\frac{1}{3} (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.4

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

$\frac{1}{3} (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 2.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{3} (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.5.2

Multiply $3$ by $- 2$ .

$\frac{1}{3} (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.6

Multiply $3$ by $- 1$ .

$\frac{1}{3} (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 2.7.1

Move $x^{2}$ .

$\frac{1}{3} (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{3} (- 3 x^{2 - 6}) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.7.3

Subtract $6$ from $2$ .

$\frac{1}{3} (- 3 x^{- 4}) + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.8

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

$\frac{- 3}{3} x^{- 4} + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.9

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

$\frac{- 3 x^{- 4}}{3} + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.10

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 3}{3 x^{4}} + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.11

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

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Step 2.11.1

Factor $3$ out of $- 3$ .

$\frac{3 \cdot - 1}{3 x^{4}} + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.11.2

Cancel the common factors.

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Step 2.11.2.1

Factor $3$ out of $3 x^{4}$ .

$\frac{3 \cdot - 1}{3 (x^{4})} + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.11.2.2

Cancel the common factor.

$\frac{3 \cdot - 1}{3 x^{4}} + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.11.2.3

Rewrite the expression.

$\frac{- 1}{x^{4}} + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 2.12

Move the negative in front of the fraction.

$- \frac{1}{x^{4}} + \frac{d}{d x} [\frac{x^{7}}{10}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{x^{7}}{10}]$ .

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Step 3.1

Since $\frac{1}{10}$ is constant with respect to $x$ , the derivative of $\frac{x^{7}}{10}$ with respect to $x$ is $\frac{1}{10} \frac{d}{d x} [x^{7}]$ .

$- \frac{1}{x^{4}} + \frac{1}{10} \frac{d}{d x} [x^{7}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$- \frac{1}{x^{4}} + \frac{1}{10} (7 x^{6})$

Step 3.3

Combine $7$ and $\frac{1}{10}$ .

$- \frac{1}{x^{4}} + \frac{7}{10} x^{6}$

Step 3.4

Combine $\frac{7}{10}$ and $x^{6}$ .

$- \frac{1}{x^{4}} + \frac{7 x^{6}}{10}$

Step 4

Reorder terms.

$\frac{7 x^{6}}{10} - \frac{1}{x^{4}}$