Calculus Examples

$f (x) = x^{4} - 10 x^{2} + 9$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{4} - 10 x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 10 x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 10 x^{2}] + \frac{d}{d x} [9]$

Step 1.1.1.2

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

$4 x^{3} + \frac{d}{d x} [- 10 x^{2}] + \frac{d}{d x} [9]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 10 x^{2}]$ .

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Step 1.1.2.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x^{2}$ with respect to $x$ is $- 10 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 10 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

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 $n = 2$ .

$4 x^{3} - 10 (2 x) + \frac{d}{d x} [9]$

Step 1.1.2.3

Multiply $2$ by $- 10$ .

$4 x^{3} - 20 x + \frac{d}{d x} [9]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$4 x^{3} - 20 x + 0$

Step 1.1.3.2

Add $4 x^{3} - 20 x$ and $0$ .

$f' (x) = 4 x^{3} - 20 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 20 x$ .

$4 x^{3} - 20 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 20 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 20 x = 0$

Step 2.2

Factor $4 x$ out of $4 x^{3} - 20 x$ .

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Step 2.2.1

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

$4 x (x^{2}) - 20 x = 0$

Step 2.2.2

Factor $4 x$ out of $- 20 x$ .

$4 x (x^{2}) + 4 x (- 5) = 0$

Step 2.2.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 5)$ .

$4 x (x^{2} - 5) = 0$

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 $0$ .

$x = 0$

$x^{2} - 5 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Step 2.5.2

Solve $x^{2} - 5 = 0$ for $x$ .

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Step 2.5.2.1

Add $5$ to both sides of the equation.

$x^{2} = 5$

Step 2.5.2.2

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

$x = \pm \sqrt{5}$

Step 2.5.2.3

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

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Step 2.5.2.3.1

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

$x = \sqrt{5}$

Step 2.5.2.3.2

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

$x = - \sqrt{5}$

Step 2.5.2.3.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 2.6

The final solution is all the values that make $4 x (x^{2} - 5) = 0$ true.

$x = 0, \sqrt{5}, - \sqrt{5}$

Step 3

The values which make the derivative equal to $0$ are $0, \sqrt{5}, - \sqrt{5}$ .

$0, \sqrt{5}, - \sqrt{5}$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - \sqrt{5}) \cup (- \sqrt{5}, 0) \cup (0, \sqrt{5}) \cup (\sqrt{5}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \sqrt{5})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.236068$ in the expression.

$f' (- 3.236068) = 4 {(- 3.236068)}^{3} - 20 \cdot - 3.236068$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 3.236068$ to the power of $3$ .

$f' (- 3.236068) = 4 \cdot - 33.88854452 - 20 \cdot - 3.236068$

Step 5.2.1.2

Multiply $4$ by $- 33.88854452$ .

$f' (- 3.236068) = - 135.5541781 - 20 \cdot - 3.236068$

Step 5.2.1.3

Multiply $- 20$ by $- 3.236068$ .

$f' (- 3.236068) = - 135.5541781 + 64.72136$

Step 5.2.2

Add $- 135.5541781$ and $64.72136$ .

$f' (- 3.236068) = - 70.8328181$

Step 5.2.3

The final answer is $- 70.8328181$ .

$- 70.8328181$

Step 5.3

At $x = - 3.236068$ the derivative is $- 70.8328181$ . Since this is negative, the function is decreasing on $(- \infty, - \sqrt{5})$ .

Decreasing on $(- \infty, - \sqrt{5})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 2.236068, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.118034$ in the expression.

$f' (- 1.118034) = 4 {(- 1.118034)}^{3} - 20 \cdot - 1.118034$

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 $- 1.118034$ to the power of $3$ .

$f' (- 1.118034) = 4 \cdot - 1.39754252 - 20 \cdot - 1.118034$

Step 6.2.1.2

Multiply $4$ by $- 1.39754252$ .

$f' (- 1.118034) = - 5.59017011 - 20 \cdot - 1.118034$

Step 6.2.1.3

Multiply $- 20$ by $- 1.118034$ .

$f' (- 1.118034) = - 5.59017011 + 22.36068$

Step 6.2.2

Add $- 5.59017011$ and $22.36068$ .

$f' (- 1.118034) = 16.77050988$

Step 6.2.3

The final answer is $16.77050988$ .

$16.77050988$

Step 6.3

At $x = - 1.118034$ the derivative is $16.77050988$ . Since this is positive, the function is increasing on $(- 2.236068, 0)$ .

Increasing on $(- \sqrt{5}, 0)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(0, \sqrt{5})$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.118034$ in the expression.

$f' (1.118034) = 4 {(1.118034)}^{3} - 20 \cdot 1.118034$

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 $1.118034$ to the power of $3$ .

$f' (1.118034) = 4 \cdot 1.39754252 - 20 \cdot 1.118034$

Step 7.2.1.2

Multiply $4$ by $1.39754252$ .

$f' (1.118034) = 5.59017011 - 20 \cdot 1.118034$

Step 7.2.1.3

Multiply $- 20$ by $1.118034$ .

$f' (1.118034) = 5.59017011 - 22.36068$

Step 7.2.2

Subtract $22.36068$ from $5.59017011$ .

$f' (1.118034) = - 16.77050988$

Step 7.2.3

The final answer is $- 16.77050988$ .

$- 16.77050988$

Step 7.3

At $x = 1.118034$ the derivative is $- 16.77050988$ . Since this is negative, the function is decreasing on $(0, \sqrt{5})$ .

Decreasing on $(0, \sqrt{5})$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(\sqrt{5}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $3.236068$ in the expression.

$f' (3.236068) = 4 {(3.236068)}^{3} - 20 \cdot 3.236068$

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 $3.236068$ to the power of $3$ .

$f' (3.236068) = 4 \cdot 33.88854452 - 20 \cdot 3.236068$

Step 8.2.1.2

Multiply $4$ by $33.88854452$ .

$f' (3.236068) = 135.5541781 - 20 \cdot 3.236068$

Step 8.2.1.3

Multiply $- 20$ by $3.236068$ .

$f' (3.236068) = 135.5541781 - 64.72136$

Step 8.2.2

Subtract $64.72136$ from $135.5541781$ .

$f' (3.236068) = 70.8328181$

Step 8.2.3

The final answer is $70.8328181$ .

$70.8328181$

Step 8.3

At $x = 3.236068$ the derivative is $70.8328181$ . Since this is positive, the function is increasing on $(\sqrt{5}, \infty)$ .

Increasing on $(\sqrt{5}, \infty)$ since $f' (x) > 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \sqrt{5}, 0), (\sqrt{5}, \infty)$

Decreasing on: $(- \infty, - \sqrt{5}), (0, \sqrt{5})$

Step 10