Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

Combine the numerators over the common denominator.

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

Step 1.1.2.5

Simplify the numerator.

Tap for more steps...

Step 1.1.2.5.1

Multiply $- 1$ by $3$ .

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

Step 1.1.2.5.2

Subtract $3$ from $1$ .

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

Step 1.1.2.6

Move the negative in front of the fraction.

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

Step 1.1.3

Evaluate $\frac{d}{d x} [x^{\frac{4}{3}}]$ .

Tap for more steps...

Step 1.1.3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

Combine the numerators over the common denominator.

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

Step 1.1.3.5

Simplify the numerator.

Tap for more steps...

Step 1.1.3.5.1

Multiply $- 1$ by $3$ .

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

Step 1.1.3.5.2

Subtract $3$ from $4$ .

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

Step 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

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

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

Step 1.1.4.2

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

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

Step 1.1.4.3

Reorder terms.

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

Step 1.1.4.4

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

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 x^{\frac{1}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{4 x^{\frac{1}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3, 3 x^{\frac{2}{3}}, 1$

Step 2.2.2

Since $3, 3 x^{\frac{2}{3}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $3, 3, 1$ then find LCM for the variable part $x^{\frac{2}{3}}$ .

Step 2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2.4

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 2.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.6

The LCM of $3, 3, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 2.2.7

The LCM of $x^{\frac{2}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

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

Step 2.2.8

The LCM for $3, 3 x^{\frac{2}{3}}, 1$ is the numeric part $3$ multiplied by the variable part.

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

Step 2.3

Multiply each term in $\frac{4 x^{\frac{1}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}} = 0$ by $3 x^{\frac{2}{3}}$ to eliminate the fractions.

Tap for more steps...

Step 2.3.1

Multiply each term in $\frac{4 x^{\frac{1}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}} = 0$ by $3 x^{\frac{2}{3}}$ .

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

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Simplify each term.

Tap for more steps...

Step 2.3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.3.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.2.1.2.1

Cancel the common factor.

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

Step 2.3.2.1.2.2

Rewrite the expression.

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

Step 2.3.2.1.3

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

Tap for more steps...

Step 2.3.2.1.3.1

Move $x^{\frac{2}{3}}$ .

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

Step 2.3.2.1.3.2

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

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

Step 2.3.2.1.3.3

Combine the numerators over the common denominator.

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

Step 2.3.2.1.3.4

Add $2$ and $1$ .

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

Step 2.3.2.1.3.5

Divide $3$ by $3$ .

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

Step 2.3.2.1.4

Simplify $4 x^{1}$ .

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

Step 2.3.2.1.5

Rewrite using the commutative property of multiplication.

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

Step 2.3.2.1.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.2.1.6.1

Cancel the common factor.

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

Step 2.3.2.1.6.2

Rewrite the expression.

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

Step 2.3.2.1.7

Cancel the common factor of $x^{\frac{2}{3}}$ .

Tap for more steps...

Step 2.3.2.1.7.1

Cancel the common factor.

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

Step 2.3.2.1.7.2

Rewrite the expression.

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

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Multiply $0 (3 x^{\frac{2}{3}})$ .

Tap for more steps...

Step 2.3.3.1.1

Multiply $3$ by $0$ .

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

Step 2.3.3.1.2

Multiply $0$ by $x^{\frac{2}{3}}$ .

$4 x + 1 = 0$

Step 2.4

Solve the equation.

Tap for more steps...

Step 2.4.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 2.4.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

Tap for more steps...

Step 2.4.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 2.4.2.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Step 2.4.2.3

Simplify the right side.

Tap for more steps...

Step 2.4.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 3.1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 3.1.3

Anything raised to $1$ is the base itself.

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

Step 3.2

Set the denominator in $\frac{1}{3 \sqrt[3]{x^{2}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{x^{2}} = 0$

Step 3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${(3 \sqrt[3]{x^{2}})}^{3} = 0^{3}$

Step 3.3.2

Simplify each side of the equation.

Tap for more steps...

Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

${(3 x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.2.1

Simplify ${(3 x^{\frac{2}{3}})}^{3}$ .

Tap for more steps...

Step 3.3.2.2.1.1

Apply the product rule to $3 x^{\frac{2}{3}}$ .

$3^{3} {(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.2

Raise $3$ to the power of $3$ .

$27 {(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

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

Tap for more steps...

Step 3.3.2.2.1.3.1

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

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{2} = 0^{3}$

Step 3.3.2.3

Simplify the right side.

Tap for more steps...

Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$27 x^{2} = 0$

Step 3.3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.3.1

Divide each term in $27 x^{2} = 0$ by $27$ and simplify.

Tap for more steps...

Step 3.3.3.1.1

Divide each term in $27 x^{2} = 0$ by $27$ .

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 3.3.3.1.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.1.2.1

Cancel the common factor of $27$ .

Tap for more steps...

Step 3.3.3.1.2.1.1

Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 3.3.3.1.2.1.2

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

$x^{2} = \frac{0}{27}$

Step 3.3.3.1.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1.3.1

Divide $0$ by $27$ .

$x^{2} = 0$

Step 3.3.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 3.3.3.3

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 3.3.3.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.3.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.3.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $x^{\frac{1}{3}} + x^{\frac{4}{3}}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = - \frac{1}{4}$ .

Tap for more steps...

Step 4.1.1

Substitute $- \frac{1}{4}$ for $x$ .

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

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.1.2.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

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

Step 4.1.2.1.1.2

Apply the product rule to $\frac{1}{4}$ .

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

Step 4.1.2.1.2

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

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

Step 4.1.2.1.3

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

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

Step 4.1.2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.1.2.1.4.1

Cancel the common factor.

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

Step 4.1.2.1.4.2

Rewrite the expression.

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

Step 4.1.2.1.5

Evaluate the exponent.

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

Step 4.1.2.1.6

One to any power is one.

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

Step 4.1.2.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.1.2.1.7.1

Apply the product rule to $- \frac{1}{4}$ .

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

Step 4.1.2.1.7.2

Apply the product rule to $\frac{1}{4}$ .

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

Step 4.1.2.1.8

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

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

Step 4.1.2.1.9

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

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

Step 4.1.2.1.10

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.1.2.1.10.1

Cancel the common factor.

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

Step 4.1.2.1.10.2

Rewrite the expression.

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

Step 4.1.2.1.11

Raise $- 1$ to the power of $4$ .

$- \frac{1}{4^{\frac{1}{3}}} + 1 \frac{1^{\frac{4}{3}}}{4^{\frac{4}{3}}}$

Step 4.1.2.1.12

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

$- \frac{1}{4^{\frac{1}{3}}} + \frac{1^{\frac{4}{3}}}{4^{\frac{4}{3}}}$

Step 4.1.2.1.13

One to any power is one.

$- \frac{1}{4^{\frac{1}{3}}} + \frac{1}{4^{\frac{4}{3}}}$

Step 4.1.2.2

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

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

Step 4.1.2.3

Write each expression with a common denominator of $4^{\frac{4}{3}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1.2.3.1

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

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

Step 4.1.2.3.2

Multiply $4^{\frac{1}{3}}$ by $4^{\frac{3}{3}}$ by adding the exponents.

Tap for more steps...

Step 4.1.2.3.2.1

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

$- \frac{4^{\frac{3}{3}}}{4^{\frac{1}{3} + \frac{3}{3}}} + \frac{1}{4^{\frac{4}{3}}}$

Step 4.1.2.3.2.2

Combine the numerators over the common denominator.

$- \frac{4^{\frac{3}{3}}}{4^{\frac{1 + 3}{3}}} + \frac{1}{4^{\frac{4}{3}}}$

Step 4.1.2.3.2.3

Add $1$ and $3$ .

$- \frac{4^{\frac{3}{3}}}{4^{\frac{4}{3}}} + \frac{1}{4^{\frac{4}{3}}}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{- 4^{\frac{3}{3}} + 1}{4^{\frac{4}{3}}}$

Step 4.1.2.5

Simplify the numerator.

Tap for more steps...

Step 4.1.2.5.1

Divide $3$ by $3$ .

$\frac{- 4^{1} + 1}{4^{\frac{4}{3}}}$

Step 4.1.2.5.2

Raise $4$ to the power of $1$ .

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

Step 4.1.2.5.3

Multiply $- 1$ by $4$ .

$\frac{- 4 + 1}{4^{\frac{4}{3}}}$

Step 4.1.2.5.4

Add $- 4$ and $1$ .

$\frac{- 3}{4^{\frac{4}{3}}}$

Step 4.1.2.6

Move the negative in front of the fraction.

$- \frac{3}{4^{\frac{4}{3}}}$

Step 4.2

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.2.1

Substitute $0$ for $x$ .

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

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

Step 4.2.2.1.1

Rewrite $0$ as $0^{3}$ .

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

Step 4.2.2.1.2

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

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

Step 4.2.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.2.1.3.1

Cancel the common factor.

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

Step 4.2.2.1.3.2

Rewrite the expression.

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

Step 4.2.2.1.4

Evaluate the exponent.

$0 + {(0)}^{\frac{4}{3}}$

Step 4.2.2.1.5

Rewrite $0$ as $0^{3}$ .

$0 + {(0^{3})}^{\frac{4}{3}}$

Step 4.2.2.1.6

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

$0 + 0^{3 (\frac{4}{3})}$

Step 4.2.2.1.7

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.2.1.7.1

Cancel the common factor.

$0 + 0^{3 (\frac{4}{3})}$

Step 4.2.2.1.7.2

Rewrite the expression.

$0 + 0^{4}$

Step 4.2.2.1.8

Raising $0$ to any positive power yields $0$ .

$0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$0$

Step 4.3

List all of the points.

$(- \frac{1}{4}, - \frac{3}{4^{\frac{4}{3}}}), (0, 0)$

Step 5