Calculus Examples

$f (x) = (x^{5} - 3 x) (\frac{1}{x^{2}})$

Step 1

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 $f (x) = x^{5} - 3 x$ and $g (x) = \frac{1}{x^{2}}$ .

$(x^{5} - 3 x) \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{1}{x^{2}} \frac{d}{d x} [x^{5} - 3 x]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$(x^{5} - 3 x) \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{1}{x^{2}} \frac{d}{d x} [x^{5} - 3 x]$

Step 2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 2.2.1

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

$(x^{5} - 3 x) \frac{d}{d x} [x^{2 \cdot - 1}] + \frac{1}{x^{2}} \frac{d}{d x} [x^{5} - 3 x]$

Step 2.2.2

Multiply $2$ by $- 1$ .

$(x^{5} - 3 x) \frac{d}{d x} [x^{- 2}] + \frac{1}{x^{2}} \frac{d}{d x} [x^{5} - 3 x]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 2$ .

$(x^{5} - 3 x) (- 2 x^{- 3}) + \frac{1}{x^{2}} \frac{d}{d x} [x^{5} - 3 x]$

Step 2.4

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

$(x^{5} - 3 x) (- 2 x^{- 3}) + \frac{1}{x^{2}} (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 3 x])$

Step 2.5

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

$(x^{5} - 3 x) (- 2 x^{- 3}) + \frac{1}{x^{2}} (5 x^{4} + \frac{d}{d x} [- 3 x])$

Step 2.6

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

$(x^{5} - 3 x) (- 2 x^{- 3}) + \frac{1}{x^{2}} (5 x^{4} - 3 \frac{d}{d x} [x])$

Step 2.7

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

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

Step 2.8

Multiply $- 3$ by $1$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Combine terms.

Tap for more steps...

Step 3.4.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

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

Step 3.4.2

Move the negative in front of the fraction.

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

Step 3.4.3

Combine $x^{5}$ and $\frac{2}{x^{3}}$ .

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

Step 3.4.4

Move $2$ to the left of $x^{5}$ .

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

Step 3.4.5

Cancel the common factor of $x^{5}$ and $x^{3}$ .

Tap for more steps...

Step 3.4.5.1

Factor $x^{3}$ out of $2 x^{5}$ .

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

Step 3.4.5.2

Cancel the common factors.

Tap for more steps...

Step 3.4.5.2.1

Multiply by $1$ .

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

Step 3.4.5.2.2

Cancel the common factor.

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

Step 3.4.5.2.3

Rewrite the expression.

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

Step 3.4.5.2.4

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

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

Step 3.4.6

Multiply $2$ by $- 1$ .

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

Step 3.4.7

Combine $- 2$ and $\frac{1}{x^{3}}$ .

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

Step 3.4.8

Move the negative in front of the fraction.

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

Step 3.4.9

Multiply $- 1$ by $- 3$ .

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

Step 3.4.10

Combine $3$ and $\frac{2}{x^{3}}$ .

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

Step 3.4.11

Multiply $3$ by $2$ .

$- 2 x^{2} + x \frac{6}{x^{3}} + \frac{1}{x^{2}} (5 x^{4}) + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.12

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

$- 2 x^{2} + \frac{x \cdot 6}{x^{3}} + \frac{1}{x^{2}} (5 x^{4}) + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.13

Move $6$ to the left of $x$ .

$- 2 x^{2} + \frac{6 \cdot x}{x^{3}} + \frac{1}{x^{2}} (5 x^{4}) + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.14

Cancel the common factor of $x$ and $x^{3}$ .

Tap for more steps...

Step 3.4.14.1

Factor $x$ out of $6 x$ .

$- 2 x^{2} + \frac{x \cdot 6}{x^{3}} + \frac{1}{x^{2}} (5 x^{4}) + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.14.2

Cancel the common factors.

Tap for more steps...

Step 3.4.14.2.1

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

$- 2 x^{2} + \frac{x \cdot 6}{x \cdot x^{2}} + \frac{1}{x^{2}} (5 x^{4}) + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.14.2.2

Cancel the common factor.

$- 2 x^{2} + \frac{x \cdot 6}{x \cdot x^{2}} + \frac{1}{x^{2}} (5 x^{4}) + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.14.2.3

Rewrite the expression.

$- 2 x^{2} + \frac{6}{x^{2}} + \frac{1}{x^{2}} (5 x^{4}) + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.15

Combine $5$ and $\frac{1}{x^{2}}$ .

$- 2 x^{2} + \frac{6}{x^{2}} + \frac{5}{x^{2}} x^{4} + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.16

Combine $\frac{5}{x^{2}}$ and $x^{4}$ .

$- 2 x^{2} + \frac{6}{x^{2}} + \frac{5 x^{4}}{x^{2}} + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.17

Cancel the common factor of $x^{4}$ and $x^{2}$ .

Tap for more steps...

Step 3.4.17.1

Factor $x^{2}$ out of $5 x^{4}$ .

$- 2 x^{2} + \frac{6}{x^{2}} + \frac{x^{2} (5 x^{2})}{x^{2}} + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.17.2

Cancel the common factors.

Tap for more steps...

Step 3.4.17.2.1

Multiply by $1$ .

$- 2 x^{2} + \frac{6}{x^{2}} + \frac{x^{2} (5 x^{2})}{x^{2} \cdot 1} + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.17.2.2

Cancel the common factor.

$- 2 x^{2} + \frac{6}{x^{2}} + \frac{x^{2} (5 x^{2})}{x^{2} \cdot 1} + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.17.2.3

Rewrite the expression.

$- 2 x^{2} + \frac{6}{x^{2}} + \frac{5 x^{2}}{1} + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.17.2.4

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

$- 2 x^{2} + \frac{6}{x^{2}} + 5 x^{2} + \frac{1}{x^{2}} \cdot - 3$

Step 3.4.18

Combine $\frac{1}{x^{2}}$ and $- 3$ .

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

Step 3.4.19

Move the negative in front of the fraction.

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

Step 3.4.20

Add $- 2 x^{2}$ and $5 x^{2}$ .

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

Step 3.4.21

Combine the numerators over the common denominator.

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

Step 3.4.22

Subtract $3$ from $6$ .

$3 x^{2} + \frac{3}{x^{2}}$