Calculus Examples

$\frac{1}{10} x^{5} - x^{4}$

Step 1

Write $\frac{1}{10} x^{5} - x^{4}$ as a function.

$f (x) = \frac{1}{10} x^{5} - x^{4}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $\frac{1}{10} x^{5} - x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{10} x^{5}] + \frac{d}{d x} [- x^{4}]$ .

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

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{1}{10} x^{5}]$ .

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

$\frac{5}{10} x^{4} + \frac{d}{d x} [- x^{4}]$

Step 2.1.2.4

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

$\frac{5 x^{4}}{10} + \frac{d}{d x} [- x^{4}]$

Step 2.1.2.5

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

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Step 2.1.2.5.1

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

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

Step 2.1.2.5.2

Cancel the common factors.

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Step 2.1.2.5.2.1

Factor $5$ out of $10$ .

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [- x^{4}]$

Step 2.1.2.5.2.2

Cancel the common factor.

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [- x^{4}]$

Step 2.1.2.5.2.3

Rewrite the expression.

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

Step 2.1.3

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

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Step 2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Multiply $4$ by $- 1$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

Step 2.2.2

Evaluate $\frac{d}{d x} [\frac{x^{4}}{2}]$ .

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Step 2.2.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{4}]$ .

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Combine $4$ and $\frac{1}{2}$ .

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

Step 2.2.2.4

Combine $\frac{4}{2}$ and $x^{3}$ .

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

Step 2.2.2.5

Cancel the common factor of $4$ and $2$ .

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Step 2.2.2.5.1

Factor $2$ out of $4 x^{3}$ .

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

Step 2.2.2.5.2

Cancel the common factors.

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Step 2.2.2.5.2.1

Factor $2$ out of $2$ .

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

Step 2.2.2.5.2.2

Cancel the common factor.

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

Step 2.2.2.5.2.3

Rewrite the expression.

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

Step 2.2.2.5.2.4

Divide $2 x^{3}$ by $1$ .

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

$2 x^{3} - 4 (3 x^{2})$

Step 2.2.3.3

Multiply $3$ by $- 4$ .

$f'' (x) = 2 x^{3} - 12 x^{2}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $2 x^{3} - 12 x^{2}$ .

$2 x^{3} - 12 x^{2}$

Step 3

Set the second derivative equal to $0$ then solve the equation $2 x^{3} - 12 x^{2} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$2 x^{3} - 12 x^{2} = 0$

Step 3.2

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

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Step 3.2.1

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

$2 x^{2} (x) - 12 x^{2} = 0$

Step 3.2.2

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

$2 x^{2} (x) + 2 x^{2} (- 6) = 0$

Step 3.2.3

Factor $2 x^{2}$ out of $2 x^{2} (x) + 2 x^{2} (- 6)$ .

$2 x^{2} (x - 6) = 0$

Step 3.3

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

$x^{2} = 0$

$x - 6 = 0$

Step 3.4

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

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Step 3.4.1

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

$x^{2} = 0$

Step 3.4.2

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

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Step 3.4.2.1

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

$x = \pm \sqrt{0}$

Step 3.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

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

$x = \pm 0$

Step 3.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.5

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.5.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.6

The final solution is all the values that make $2 x^{2} (x - 6) = 0$ true.

$x = 0, 6$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $0$ in $f (x) = \frac{1}{10} x^{5} - x^{4}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{1}{10} \cdot {(0)}^{5} - {(0)}^{4}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$f (0) = \frac{1}{10} \cdot 0 - {(0)}^{4}$

Step 4.1.2.1.2

Multiply $\frac{1}{10}$ by $0$ .

$f (0) = 0 - {(0)}^{4}$

Step 4.1.2.1.3

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

$f (0) = 0 - 0$

Step 4.1.2.1.4

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0$

Step 4.1.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 4.1.2.3

The final answer is $0$ .

$0$

Step 4.2

The point found by substituting $0$ in $f (x) = \frac{1}{10} x^{5} - x^{4}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $6$ in $f (x) = \frac{1}{10} x^{5} - x^{4}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{1}{10} \cdot {(6)}^{5} - {(6)}^{4}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $6$ to the power of $5$ .

$f (6) = \frac{1}{10} \cdot 7776 - {(6)}^{4}$

Step 4.3.2.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.2.1

Factor $2$ out of $10$ .

$f (6) = \frac{1}{2 (5)} \cdot 7776 - {(6)}^{4}$

Step 4.3.2.1.2.2

Factor $2$ out of $7776$ .

$f (6) = \frac{1}{2 \cdot 5} \cdot (2 \cdot 3888) - {(6)}^{4}$

Step 4.3.2.1.2.3

Cancel the common factor.

$f (6) = \frac{1}{2 \cdot 5} \cdot (2 \cdot 3888) - {(6)}^{4}$

Step 4.3.2.1.2.4

Rewrite the expression.

$f (6) = \frac{1}{5} \cdot 3888 - {(6)}^{4}$

Step 4.3.2.1.3

Combine $\frac{1}{5}$ and $3888$ .

$f (6) = \frac{3888}{5} - {(6)}^{4}$

Step 4.3.2.1.4

Raise $6$ to the power of $4$ .

$f (6) = \frac{3888}{5} - 1 \cdot 1296$

Step 4.3.2.1.5

Multiply $- 1$ by $1296$ .

$f (6) = \frac{3888}{5} - 1296$

Step 4.3.2.2

To write $- 1296$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (6) = \frac{3888}{5} - 1296 \cdot \frac{5}{5}$

Step 4.3.2.3

Combine $- 1296$ and $\frac{5}{5}$ .

$f (6) = \frac{3888}{5} + \frac{- 1296 \cdot 5}{5}$

Step 4.3.2.4

Combine the numerators over the common denominator.

$f (6) = \frac{3888 - 1296 \cdot 5}{5}$

Step 4.3.2.5

Simplify the numerator.

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Step 4.3.2.5.1

Multiply $- 1296$ by $5$ .

$f (6) = \frac{3888 - 6480}{5}$

Step 4.3.2.5.2

Subtract $6480$ from $3888$ .

$f (6) = \frac{- 2592}{5}$

Step 4.3.2.6

Move the negative in front of the fraction.

$f (6) = - \frac{2592}{5}$

Step 4.3.2.7

The final answer is $- \frac{2592}{5}$ .

$- \frac{2592}{5}$

Step 4.4

The point found by substituting $6$ in $f (x) = \frac{1}{10} x^{5} - x^{4}$ is $(6, - \frac{2592}{5})$ . This point can be an inflection point.

$(6, - \frac{2592}{5})$

Step 4.5

Determine the points that could be inflection points.

$(0, 0), (6, - \frac{2592}{5})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, 0) \cup (0, 6) \cup (6, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.1$ in the expression.

$f'' (- 0.1) = 2 {(- 0.1)}^{3} - 12 {(- 0.1)}^{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 0.1$ to the power of $3$ .

$f'' (- 0.1) = 2 \cdot - 0.001 - 12 {(- 0.1)}^{2}$

Step 6.2.1.2

Multiply $2$ by $- 0.001$ .

$f'' (- 0.1) = - 0.002 - 12 {(- 0.1)}^{2}$

Step 6.2.1.3

Raise $- 0.1$ to the power of $2$ .

$f'' (- 0.1) = - 0.002 - 12 \cdot 0.01$

Step 6.2.1.4

Multiply $- 12$ by $0.01$ .

$f'' (- 0.1) = - 0.002 - 0.12$

Step 6.2.2

Subtract $0.12$ from $- 0.002$ .

$f'' (- 0.1) = - 0.122$

Step 6.2.3

The final answer is $- 0.122$ .

$- 0.122$

Step 6.3

At $- 0.1$ , the second derivative is $- 0.122$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, 0)$

Decreasing on $(- \infty, 0)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0, 6)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3$ in the expression.

$f'' (3) = 2 {(3)}^{3} - 12 {(3)}^{2}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

$f'' (3) = 2 \cdot 27 - 12 {(3)}^{2}$

Step 7.2.1.2

Multiply $2$ by $27$ .

$f'' (3) = 54 - 12 {(3)}^{2}$

Step 7.2.1.3

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

$f'' (3) = 54 - 12 \cdot 9$

Step 7.2.1.4

Multiply $- 12$ by $9$ .

$f'' (3) = 54 - 108$

Step 7.2.2

Subtract $108$ from $54$ .

$f'' (3) = - 54$

Step 7.2.3

The final answer is $- 54$ .

$- 54$

Step 7.3

At $3$ , the second derivative is $- 54$ . Since this is negative, the second derivative is decreasing on the interval $(0, 6)$

Decreasing on $(0, 6)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(6, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $6.1$ in the expression.

$f'' (6.1) = 2 {(6.1)}^{3} - 12 {(6.1)}^{2}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $6.1$ to the power of $3$ .

$f'' (6.1) = 2 \cdot 226.981 - 12 {(6.1)}^{2}$

Step 8.2.1.2

Multiply $2$ by $226.981$ .

$f'' (6.1) = 453.962 - 12 {(6.1)}^{2}$

Step 8.2.1.3

Raise $6.1$ to the power of $2$ .

$f'' (6.1) = 453.962 - 12 \cdot 37.21$

Step 8.2.1.4

Multiply $- 12$ by $37.21$ .

$f'' (6.1) = 453.962 - 446.52$

Step 8.2.2

Subtract $446.52$ from $453.962$ .

$f'' (6.1) = 7.442$

Step 8.2.3

The final answer is $7.442$ .

$7.442$

Step 8.3

At $6.1$ , the second derivative is $7.442$ . Since this is positive, the second derivative is increasing on the interval $(6, \infty)$ .

Increasing on $(6, \infty)$ since $f'' (x) > 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(6, - \frac{2592}{5})$ .

$(6, - \frac{2592}{5})$

Step 10