Calculus Examples

$\frac{1}{5} x^{5} - \frac{4}{3} x^{3} + \frac{17}{15}$

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 $\frac{1}{5} x^{5} - \frac{4}{3} x^{3} + \frac{17}{15}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [- \frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{17}{15}]$ .

$\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [- \frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{5} x^{5}]$ .

Tap for more steps...

Step 1.1.2.1

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

$\frac{1}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- \frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.2.2

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

$\frac{1}{5} (5 x^{4}) + \frac{d}{d x} [- \frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.2.3

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

$\frac{5}{5} x^{4} + \frac{d}{d x} [- \frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.2.4

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

$\frac{5 x^{4}}{5} + \frac{d}{d x} [- \frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.2.5

Cancel the common factor of $5$ .

Tap for more steps...

Step 1.1.2.5.1

Cancel the common factor.

$\frac{5 x^{4}}{5} + \frac{d}{d x} [- \frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.2.5.2

Divide $x^{4}$ by $1$ .

$x^{4} + \frac{d}{d x} [- \frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

Since $- \frac{4}{3}$ is constant with respect to $x$ , the derivative of $- \frac{4}{3} x^{3}$ with respect to $x$ is $- \frac{4}{3} \frac{d}{d x} [x^{3}]$ .

$x^{4} - \frac{4}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.2

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

$x^{4} - \frac{4}{3} (3 x^{2}) + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.3

Multiply $3$ by $- 1$ .

$x^{4} - 3 (\frac{4}{3}) x^{2} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.4

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

$x^{4} + \frac{- 3 \cdot 4}{3} x^{2} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.5

Multiply $- 3$ by $4$ .

$x^{4} + \frac{- 12}{3} x^{2} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.6

Combine $\frac{- 12}{3}$ and $x^{2}$ .

$x^{4} + \frac{- 12 x^{2}}{3} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.7

Cancel the common factor of $- 12$ and $3$ .

Tap for more steps...

Step 1.1.3.7.1

Factor $3$ out of $- 12 x^{2}$ .

$x^{4} + \frac{3 (- 4 x^{2})}{3} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.7.2

Cancel the common factors.

Tap for more steps...

Step 1.1.3.7.2.1

Factor $3$ out of $3$ .

$x^{4} + \frac{3 (- 4 x^{2})}{3 (1)} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.7.2.2

Cancel the common factor.

$x^{4} + \frac{3 (- 4 x^{2})}{3 \cdot 1} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.7.2.3

Rewrite the expression.

$x^{4} + \frac{- 4 x^{2}}{1} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.3.7.2.4

Divide $- 4 x^{2}$ by $1$ .

$x^{4} - 4 x^{2} + \frac{d}{d x} [\frac{17}{15}]$

Step 1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.4.1

Since $\frac{17}{15}$ is constant with respect to $x$ , the derivative of $\frac{17}{15}$ with respect to $x$ is $0$ .

$x^{4} - 4 x^{2} + 0$

Step 1.1.4.2

Add $x^{4} - 4 x^{2}$ and $0$ .

$f' (x) = x^{4} - 4 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{4} - 4 x^{2}$ .

$x^{4} - 4 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $x^{4} - 4 x^{2} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$x^{4} - 4 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

Tap for more steps...

Step 2.2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - 4 x^{2} = 0$

Step 2.2.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - 4 u = 0$

Step 2.2.3

Factor $u$ out of $u^{2} - 4 u$ .

Tap for more steps...

Step 2.2.3.1

Factor $u$ out of $u^{2}$ .

$u \cdot u - 4 u = 0$

Step 2.2.3.2

Factor $u$ out of $- 4 u$ .

$u \cdot u + u \cdot - 4 = 0$

Step 2.2.3.3

Factor $u$ out of $u \cdot u + u \cdot - 4$ .

$u (u - 4) = 0$

Step 2.2.4

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} (x^{2} - 4) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x^{2} - 4 = 0$

Step 2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

Tap for more steps...

Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x^{2} - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $x^{2} - 4$ equal to $0$ .

$x^{2} - 4 = 0$

Step 2.5.2

Solve $x^{2} - 4 = 0$ for $x$ .

Tap for more steps...

Step 2.5.2.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 2.5.2.2

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

$x = \pm \sqrt{4}$

Step 2.5.2.3

Simplify $\pm \sqrt{4}$ .

Tap for more steps...

Step 2.5.2.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.5.2.3.2

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

$x = \pm 2$

Step 2.5.2.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 2.5.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 2.5.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 2.6

The final solution is all the values that make $x^{2} (x^{2} - 4) = 0$ true.

$x = 0, 2, - 2$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{1}{5} x^{5} - \frac{4}{3} x^{3} + \frac{17}{15}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

$\frac{1}{5} \cdot {(0)}^{5} - \frac{4}{3} \cdot {(0)}^{3} + \frac{17}{15}$

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

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

$\frac{1}{5} \cdot 0 - \frac{4}{3} \cdot {(0)}^{3} + \frac{17}{15}$

Step 4.1.2.1.2

Multiply $\frac{1}{5}$ by $0$ .

$0 - \frac{4}{3} \cdot {(0)}^{3} + \frac{17}{15}$

Step 4.1.2.1.3

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

$0 - \frac{4}{3} \cdot 0 + \frac{17}{15}$

Step 4.1.2.1.4

Multiply $- \frac{4}{3} \cdot 0$ .

Tap for more steps...

Step 4.1.2.1.4.1

Multiply $0$ by $- 1$ .

$0 + 0 (\frac{4}{3}) + \frac{17}{15}$

Step 4.1.2.1.4.2

Multiply $0$ by $\frac{4}{3}$ .

$0 + 0 + \frac{17}{15}$

Step 4.1.2.2

Simplify by adding numbers.

Tap for more steps...

Step 4.1.2.2.1

Add $0$ and $0$ .

$0 + \frac{17}{15}$

Step 4.1.2.2.2

Add $0$ and $\frac{17}{15}$ .

$\frac{17}{15}$

Step 4.2

Evaluate at $x = 2$ .

Tap for more steps...

Step 4.2.1

Substitute $2$ for $x$ .

$\frac{1}{5} \cdot {(2)}^{5} - \frac{4}{3} \cdot {(2)}^{3} + \frac{17}{15}$

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

Raise $2$ to the power of $5$ .

$\frac{1}{5} \cdot 32 - \frac{4}{3} \cdot {(2)}^{3} + \frac{17}{15}$

Step 4.2.2.1.2

Combine $\frac{1}{5}$ and $32$ .

$\frac{32}{5} - \frac{4}{3} \cdot {(2)}^{3} + \frac{17}{15}$

Step 4.2.2.1.3

Raise $2$ to the power of $3$ .

$\frac{32}{5} - \frac{4}{3} \cdot 8 + \frac{17}{15}$

Step 4.2.2.1.4

Multiply $- \frac{4}{3} \cdot 8$ .

Tap for more steps...

Step 4.2.2.1.4.1

Multiply $8$ by $- 1$ .

$\frac{32}{5} - 8 (\frac{4}{3}) + \frac{17}{15}$

Step 4.2.2.1.4.2

Combine $- 8$ and $\frac{4}{3}$ .

$\frac{32}{5} + \frac{- 8 \cdot 4}{3} + \frac{17}{15}$

Step 4.2.2.1.4.3

Multiply $- 8$ by $4$ .

$\frac{32}{5} + \frac{- 32}{3} + \frac{17}{15}$

Step 4.2.2.1.5

Move the negative in front of the fraction.

$\frac{32}{5} - \frac{32}{3} + \frac{17}{15}$

Step 4.2.2.2

Find the common denominator.

Tap for more steps...

Step 4.2.2.2.1

Multiply $\frac{32}{5}$ by $\frac{3}{3}$ .

$\frac{32}{5} \cdot \frac{3}{3} - \frac{32}{3} + \frac{17}{15}$

Step 4.2.2.2.2

Multiply $\frac{32}{5}$ by $\frac{3}{3}$ .

$\frac{32 \cdot 3}{5 \cdot 3} - \frac{32}{3} + \frac{17}{15}$

Step 4.2.2.2.3

Multiply $\frac{32}{3}$ by $\frac{5}{5}$ .

$\frac{32 \cdot 3}{5 \cdot 3} - (\frac{32}{3} \cdot \frac{5}{5}) + \frac{17}{15}$

Step 4.2.2.2.4

Multiply $\frac{32}{3}$ by $\frac{5}{5}$ .

$\frac{32 \cdot 3}{5 \cdot 3} - \frac{32 \cdot 5}{3 \cdot 5} + \frac{17}{15}$

Step 4.2.2.2.5

Reorder the factors of $5 \cdot 3$ .

$\frac{32 \cdot 3}{3 \cdot 5} - \frac{32 \cdot 5}{3 \cdot 5} + \frac{17}{15}$

Step 4.2.2.2.6

Multiply $3$ by $5$ .

$\frac{32 \cdot 3}{15} - \frac{32 \cdot 5}{3 \cdot 5} + \frac{17}{15}$

Step 4.2.2.2.7

Multiply $3$ by $5$ .

$\frac{32 \cdot 3}{15} - \frac{32 \cdot 5}{15} + \frac{17}{15}$

Step 4.2.2.3

Combine the numerators over the common denominator.

$\frac{32 \cdot 3 - 32 \cdot 5 + 17}{15}$

Step 4.2.2.4

Simplify each term.

Tap for more steps...

Step 4.2.2.4.1

Multiply $32$ by $3$ .

$\frac{96 - 32 \cdot 5 + 17}{15}$

Step 4.2.2.4.2

Multiply $- 32$ by $5$ .

$\frac{96 - 160 + 17}{15}$

Step 4.2.2.5

Simplify the expression.

Tap for more steps...

Step 4.2.2.5.1

Subtract $160$ from $96$ .

$\frac{- 64 + 17}{15}$

Step 4.2.2.5.2

Add $- 64$ and $17$ .

$\frac{- 47}{15}$

Step 4.2.2.5.3

Move the negative in front of the fraction.

$- \frac{47}{15}$

Step 4.3

Evaluate at $x = - 2$ .

Tap for more steps...

Step 4.3.1

Substitute $- 2$ for $x$ .

$\frac{1}{5} \cdot {(- 2)}^{5} - \frac{4}{3} \cdot {(- 2)}^{3} + \frac{17}{15}$

Step 4.3.2

Simplify.

Tap for more steps...

Step 4.3.2.1

Simplify each term.

Tap for more steps...

Step 4.3.2.1.1

Raise $- 2$ to the power of $5$ .

$\frac{1}{5} \cdot - 32 - \frac{4}{3} \cdot {(- 2)}^{3} + \frac{17}{15}$

Step 4.3.2.1.2

Combine $\frac{1}{5}$ and $- 32$ .

$\frac{- 32}{5} - \frac{4}{3} \cdot {(- 2)}^{3} + \frac{17}{15}$

Step 4.3.2.1.3

Move the negative in front of the fraction.

$- \frac{32}{5} - \frac{4}{3} \cdot {(- 2)}^{3} + \frac{17}{15}$

Step 4.3.2.1.4

Raise $- 2$ to the power of $3$ .

$- \frac{32}{5} - \frac{4}{3} \cdot - 8 + \frac{17}{15}$

Step 4.3.2.1.5

Multiply $- \frac{4}{3} \cdot - 8$ .

Tap for more steps...

Step 4.3.2.1.5.1

Multiply $- 8$ by $- 1$ .

$- \frac{32}{5} + 8 (\frac{4}{3}) + \frac{17}{15}$

Step 4.3.2.1.5.2

Combine $8$ and $\frac{4}{3}$ .

$- \frac{32}{5} + \frac{8 \cdot 4}{3} + \frac{17}{15}$

Step 4.3.2.1.5.3

Multiply $8$ by $4$ .

$- \frac{32}{5} + \frac{32}{3} + \frac{17}{15}$

Step 4.3.2.2

Find the common denominator.

Tap for more steps...

Step 4.3.2.2.1

Multiply $\frac{32}{5}$ by $\frac{3}{3}$ .

$- (\frac{32}{5} \cdot \frac{3}{3}) + \frac{32}{3} + \frac{17}{15}$

Step 4.3.2.2.2

Multiply $\frac{32}{5}$ by $\frac{3}{3}$ .

$- \frac{32 \cdot 3}{5 \cdot 3} + \frac{32}{3} + \frac{17}{15}$

Step 4.3.2.2.3

Multiply $\frac{32}{3}$ by $\frac{5}{5}$ .

$- \frac{32 \cdot 3}{5 \cdot 3} + \frac{32}{3} \cdot \frac{5}{5} + \frac{17}{15}$

Step 4.3.2.2.4

Multiply $\frac{32}{3}$ by $\frac{5}{5}$ .

$- \frac{32 \cdot 3}{5 \cdot 3} + \frac{32 \cdot 5}{3 \cdot 5} + \frac{17}{15}$

Step 4.3.2.2.5

Reorder the factors of $5 \cdot 3$ .

$- \frac{32 \cdot 3}{3 \cdot 5} + \frac{32 \cdot 5}{3 \cdot 5} + \frac{17}{15}$

Step 4.3.2.2.6

Multiply $3$ by $5$ .

$- \frac{32 \cdot 3}{15} + \frac{32 \cdot 5}{3 \cdot 5} + \frac{17}{15}$

Step 4.3.2.2.7

Multiply $3$ by $5$ .

$- \frac{32 \cdot 3}{15} + \frac{32 \cdot 5}{15} + \frac{17}{15}$

Step 4.3.2.3

Combine the numerators over the common denominator.

$\frac{- 32 \cdot 3 + 32 \cdot 5 + 17}{15}$

Step 4.3.2.4

Simplify each term.

Tap for more steps...

Step 4.3.2.4.1

Multiply $- 32$ by $3$ .

$\frac{- 96 + 32 \cdot 5 + 17}{15}$

Step 4.3.2.4.2

Multiply $32$ by $5$ .

$\frac{- 96 + 160 + 17}{15}$

Step 4.3.2.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.3.2.5.1

Add $- 96$ and $160$ .

$\frac{64 + 17}{15}$

Step 4.3.2.5.2

Add $64$ and $17$ .

$\frac{81}{15}$

Step 4.3.2.5.3

Cancel the common factor of $81$ and $15$ .

Tap for more steps...

Step 4.3.2.5.3.1

Factor $3$ out of $81$ .

$\frac{3 (27)}{15}$

Step 4.3.2.5.3.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.5.3.2.1

Factor $3$ out of $15$ .

$\frac{3 \cdot 27}{3 \cdot 5}$

Step 4.3.2.5.3.2.2

Cancel the common factor.

$\frac{3 \cdot 27}{3 \cdot 5}$

Step 4.3.2.5.3.2.3

Rewrite the expression.

$\frac{27}{5}$

Step 4.4

List all of the points.

$(0, \frac{17}{15}), (2, - \frac{47}{15}), (- 2, \frac{27}{5})$

Step 5