Calculus Examples

$y = \frac{x^{5}}{10} + \frac{1}{6 x^{3}}$

Step 1

By the Sum Rule, the derivative of $\frac{x^{5}}{10} + \frac{1}{6 x^{3}}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{5}}{10}] + \frac{d}{d x} [\frac{1}{6 x^{3}}]$ .

$\frac{d}{d x} [\frac{x^{5}}{10}] + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{x^{5}}{10}]$ .

Tap for more steps...

Step 2.1

Since $\frac{1}{10}$ is constant with respect to $x$ , the derivative of $\frac{x^{5}}{10}$ with respect to $x$ is $\frac{1}{10} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{10} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 2.2

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

$\frac{1}{10} (5 x^{4}) + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 2.3

Combine $5$ and $\frac{1}{10}$ .

$\frac{5}{10} x^{4} + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 2.4

Combine $\frac{5}{10}$ and $x^{4}$ .

$\frac{5 x^{4}}{10} + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 2.5

Cancel the common factor of $5$ and $10$ .

Tap for more steps...

Step 2.5.1

Factor $5$ out of $5 x^{4}$ .

$\frac{5 (x^{4})}{10} + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 2.5.2

Cancel the common factors.

Tap for more steps...

Step 2.5.2.1

Factor $5$ out of $10$ .

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 2.5.2.2

Cancel the common factor.

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 2.5.2.3

Rewrite the expression.

$\frac{x^{4}}{2} + \frac{d}{d x} [\frac{1}{6 x^{3}}]$

Step 3

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

Tap for more steps...

Step 3.1

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

$\frac{x^{4}}{2} + \frac{1}{6} \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.2

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

$\frac{x^{4}}{2} + \frac{1}{6} \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 3.3

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 $g (x) = x^{3}$ .

Tap for more steps...

Step 3.3.1

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

$\frac{x^{4}}{2} + \frac{1}{6} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}])$

Step 3.3.2

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

$\frac{x^{4}}{2} + \frac{1}{6} (- u^{- 2} \frac{d}{d x} [x^{3}])$

Step 3.3.3

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

$\frac{x^{4}}{2} + \frac{1}{6} (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}])$

Step 3.4

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

$\frac{x^{4}}{2} + \frac{1}{6} (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$\frac{x^{4}}{2} + \frac{1}{6} (- x^{3 \cdot - 2} (3 x^{2}))$

Step 3.5.2

Multiply $3$ by $- 2$ .

$\frac{x^{4}}{2} + \frac{1}{6} (- x^{- 6} (3 x^{2}))$

Step 3.6

Multiply $3$ by $- 1$ .

$\frac{x^{4}}{2} + \frac{1}{6} (- 3 x^{- 6} x^{2})$

Step 3.7

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

Tap for more steps...

Step 3.7.1

Move $x^{2}$ .

$\frac{x^{4}}{2} + \frac{1}{6} (- 3 (x^{2} x^{- 6}))$

Step 3.7.2

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

$\frac{x^{4}}{2} + \frac{1}{6} (- 3 x^{2 - 6})$

Step 3.7.3

Subtract $6$ from $2$ .

$\frac{x^{4}}{2} + \frac{1}{6} (- 3 x^{- 4})$

Step 3.8

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

$\frac{x^{4}}{2} + \frac{- 3}{6} x^{- 4}$

Step 3.9

Combine $\frac{- 3}{6}$ and $x^{- 4}$ .

$\frac{x^{4}}{2} + \frac{- 3 x^{- 4}}{6}$

Step 3.10

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

$\frac{x^{4}}{2} + \frac{- 3}{6 x^{4}}$

Step 3.11

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

Tap for more steps...

Step 3.11.1

Factor $3$ out of $- 3$ .

$\frac{x^{4}}{2} + \frac{3 (- 1)}{6 x^{4}}$

Step 3.11.2

Cancel the common factors.

Tap for more steps...

Step 3.11.2.1

Factor $3$ out of $6 x^{4}$ .

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

Step 3.11.2.2

Cancel the common factor.

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

Step 3.11.2.3

Rewrite the expression.

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

Step 3.12

Move the negative in front of the fraction.

$\frac{x^{4}}{2} - \frac{1}{2 x^{4}}$