Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{2 x^{2} - 3 x + 1}{x} + x^{\frac{2}{3}} - x^{\frac{1}{3}} + 4$ with respect to $x$ is $\frac{d}{d x} [\frac{2 x^{2} - 3 x + 1}{x}] + \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x^{\frac{1}{3}}] + \frac{d}{d x} [4]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\frac{2 x^{2} - 3 x + 1}{x}]$ .

Tap for more steps...

Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 2 x^{2} - 3 x + 1$ and $g (x) = x$ .

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

Step 2.2

By the Sum Rule, the derivative of $2 x^{2} - 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$ .

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

Step 2.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x^{2}]$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{x (2 (2 x) - 3 \cdot 1 + 0) - (2 x^{2} - 3 x + 1) \frac{d}{d x} [x]}{x^{2}} + \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x^{\frac{1}{3}}] + \frac{d}{d x} [4]$

Step 2.8

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

$\frac{x (2 (2 x) - 3 \cdot 1 + 0) - (2 x^{2} - 3 x + 1) \cdot 1}{x^{2}} + \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x^{\frac{1}{3}}] + \frac{d}{d x} [4]$

Step 2.9

Multiply $2$ by $2$ .

$\frac{x (4 x - 3 \cdot 1 + 0) - (2 x^{2} - 3 x + 1) \cdot 1}{x^{2}} + \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x^{\frac{1}{3}}] + \frac{d}{d x} [4]$

Step 2.10

Multiply $- 3$ by $1$ .

$\frac{x (4 x - 3 + 0) - (2 x^{2} - 3 x + 1) \cdot 1}{x^{2}} + \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x^{\frac{1}{3}}] + \frac{d}{d x} [4]$

Step 2.11

Add $4 x - 3$ and $0$ .

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

Step 2.12

Multiply $- 1$ by $1$ .

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

Step 3

Evaluate $\frac{d}{d x} [x^{\frac{2}{3}}]$ .

Tap for more steps...

Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

Tap for more steps...

Step 3.5.1

Multiply $- 1$ by $3$ .

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

Step 3.5.2

Subtract $3$ from $2$ .

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

Tap for more steps...

Step 4.6.1

Multiply $- 1$ by $3$ .

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

Step 4.6.2

Subtract $3$ from $1$ .

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

Step 4.7

Move the negative in front of the fraction.

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

Step 4.8

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

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

Step 4.9

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x (4 x - 3) - (2 x^{2} - 3 x + 1)}{x^{2}} + \frac{2}{3} x^{- \frac{1}{3}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

$\frac{x (4 x - 3) - (2 x^{2} - 3 x + 1)}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.2

Apply the distributive property.

$\frac{x (4 x) + x \cdot - 3 - (2 x^{2} - 3 x + 1)}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.3

Apply the distributive property.

$\frac{x (4 x) + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4

Combine terms.

Tap for more steps...

Step 6.4.1

Raise $x$ to the power of $1$ .

$\frac{4 (x^{1} x) + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.2

Raise $x$ to the power of $1$ .

$\frac{4 (x^{1} x^{1}) + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.3

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

$\frac{4 x^{1 + 1} + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.4

Add $1$ and $1$ .

$\frac{4 x^{2} + x \cdot - 3 - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.5

Move $- 3$ to the left of $x$ .

$\frac{4 x^{2} - 3 \cdot x - (2 x^{2}) - (- 3 x) - 1 \cdot 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.6

Multiply $2$ by $- 1$ .

$\frac{4 x^{2} - 3 x - 2 x^{2} - (- 3 x) - 1 \cdot 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.7

Multiply $- 3$ by $- 1$ .

$\frac{4 x^{2} - 3 x - 2 x^{2} + 3 x - 1 \cdot 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.8

Multiply $- 1$ by $1$ .

$\frac{4 x^{2} - 3 x - 2 x^{2} + 3 x - 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.9

Subtract $2 x^{2}$ from $4 x^{2}$ .

$\frac{2 x^{2} - 3 x + 3 x - 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.10

Add $- 3 x$ and $3 x$ .

$\frac{2 x^{2} + 0 - 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.11

Add $2 x^{2}$ and $0$ .

$\frac{2 x^{2} - 1}{x^{2}} + \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.12

Multiply $\frac{2}{3}$ by $\frac{1}{x^{\frac{1}{3}}}$ .

$\frac{2 x^{2} - 1}{x^{2}} + \frac{2}{3 x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.13

To write $\frac{2 x^{2} - 1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 x^{2} - 1}{x^{2}} \cdot \frac{3}{3} + \frac{2}{3 x^{\frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.14

To write $\frac{2}{3 x^{\frac{1}{3}}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{5}{3}}}{x^{\frac{5}{3}}}$ .

$\frac{2 x^{2} - 1}{x^{2}} \cdot \frac{3}{3} + \frac{2}{3 x^{\frac{1}{3}}} \cdot \frac{x^{\frac{5}{3}}}{x^{\frac{5}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.15

Write each expression with a common denominator of $3 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 6.4.15.1

Multiply $\frac{2 x^{2} - 1}{x^{2}}$ by $\frac{3}{3}$ .

$\frac{(2 x^{2} - 1) \cdot 3}{x^{2} \cdot 3} + \frac{2}{3 x^{\frac{1}{3}}} \cdot \frac{x^{\frac{5}{3}}}{x^{\frac{5}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.15.2

Multiply $\frac{2}{3 x^{\frac{1}{3}}}$ by $\frac{x^{\frac{5}{3}}}{x^{\frac{5}{3}}}$ .

$\frac{(2 x^{2} - 1) \cdot 3}{x^{2} \cdot 3} + \frac{2 x^{\frac{5}{3}}}{3 x^{\frac{1}{3}} x^{\frac{5}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.15.3

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{5}{3}}$ by adding the exponents.

Tap for more steps...

Step 6.4.15.3.1

Move $x^{\frac{5}{3}}$ .

$\frac{(2 x^{2} - 1) \cdot 3}{x^{2} \cdot 3} + \frac{2 x^{\frac{5}{3}}}{3 (x^{\frac{5}{3}} x^{\frac{1}{3}})} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.15.3.2

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

$\frac{(2 x^{2} - 1) \cdot 3}{x^{2} \cdot 3} + \frac{2 x^{\frac{5}{3}}}{3 x^{\frac{5}{3} + \frac{1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.15.3.3

Combine the numerators over the common denominator.

$\frac{(2 x^{2} - 1) \cdot 3}{x^{2} \cdot 3} + \frac{2 x^{\frac{5}{3}}}{3 x^{\frac{5 + 1}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.15.3.4

Add $5$ and $1$ .

$\frac{(2 x^{2} - 1) \cdot 3}{x^{2} \cdot 3} + \frac{2 x^{\frac{5}{3}}}{3 x^{\frac{6}{3}}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.15.3.5

Divide $6$ by $3$ .

$\frac{(2 x^{2} - 1) \cdot 3}{x^{2} \cdot 3} + \frac{2 x^{\frac{5}{3}}}{3 x^{2}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.15.4

Reorder the factors of $x^{2} \cdot 3$ .

$\frac{(2 x^{2} - 1) \cdot 3}{3 x^{2}} + \frac{2 x^{\frac{5}{3}}}{3 x^{2}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.16

Combine the numerators over the common denominator.

$\frac{(2 x^{2} - 1) \cdot 3 + 2 x^{\frac{5}{3}}}{3 x^{2}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.17

Move $3$ to the left of $2 x^{2} - 1$ .

$\frac{3 \cdot (2 x^{2} - 1) + 2 x^{\frac{5}{3}}}{3 x^{2}} - \frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 6.4.18

Add $\frac{3 (2 x^{2} - 1) + 2 x^{\frac{5}{3}}}{3 x^{2}} - \frac{1}{3 x^{\frac{2}{3}}}$ and $0$ .

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

Step 6.5

Simplify the numerator.

Tap for more steps...

Step 6.5.1

Apply the distributive property.

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

Step 6.5.2

Multiply $2$ by $3$ .

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

Step 6.5.3

Multiply $3$ by $- 1$ .

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

Step 6.5.4

Reorder terms.

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

Step 6.6

To write $- \frac{1}{3 x^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{4}{3}}}{x^{\frac{4}{3}}}$ .

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

Step 6.7

Write each expression with a common denominator of $3 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 6.7.1

Multiply $\frac{1}{3 x^{\frac{2}{3}}}$ by $\frac{x^{\frac{4}{3}}}{x^{\frac{4}{3}}}$ .

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

Step 6.7.2

Multiply $x^{\frac{2}{3}}$ by $x^{\frac{4}{3}}$ by adding the exponents.

Tap for more steps...

Step 6.7.2.1

Move $x^{\frac{4}{3}}$ .

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

Step 6.7.2.2

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

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

Step 6.7.2.3

Combine the numerators over the common denominator.

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

Step 6.7.2.4

Add $4$ and $2$ .

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

Step 6.7.2.5

Divide $6$ by $3$ .

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

Step 6.8

Combine the numerators over the common denominator.

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

Step 6.9

Reorder terms.

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