Calculus Examples

$f (x) = | 25 x^{2} - 4 |$

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 using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 25 x^{2} - 4$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $25 x^{2} - 4$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [25 x^{2} - 4]$

Step 1.1.1.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$\frac{u}{| u |} \frac{d}{d x} [25 x^{2} - 4]$

Step 1.1.1.3

Replace all occurrences of $u$ with $25 x^{2} - 4$ .

$\frac{25 x^{2} - 4}{| 25 x^{2} - 4 |} \frac{d}{d x} [25 x^{2} - 4]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $25 x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{25 x^{2} - 4}{| 25 x^{2} - 4 |} (\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [- 4])$

Step 1.1.2.2

Since $25$ is constant with respect to $x$ , the derivative of $25 x^{2}$ with respect to $x$ is $25 \frac{d}{d x} [x^{2}]$ .

$\frac{25 x^{2} - 4}{| 25 x^{2} - 4 |} (25 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4])$

Step 1.1.2.3

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

$\frac{25 x^{2} - 4}{| 25 x^{2} - 4 |} (25 (2 x) + \frac{d}{d x} [- 4])$

Step 1.1.2.4

Multiply $2$ by $25$ .

$\frac{25 x^{2} - 4}{| 25 x^{2} - 4 |} (50 x + \frac{d}{d x} [- 4])$

Step 1.1.2.5

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

$\frac{25 x^{2} - 4}{| 25 x^{2} - 4 |} (50 x + 0)$

Step 1.1.2.6

Combine fractions.

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Step 1.1.2.6.1

Add $50 x$ and $0$ .

$\frac{25 x^{2} - 4}{| 25 x^{2} - 4 |} (50 x)$

Step 1.1.2.6.2

Combine $50$ and $\frac{25 x^{2} - 4}{| 25 x^{2} - 4 |}$ .

$\frac{50 (25 x^{2} - 4)}{| 25 x^{2} - 4 |} x$

Step 1.1.2.6.3

Combine $\frac{50 (25 x^{2} - 4)}{| 25 x^{2} - 4 |}$ and $x$ .

$\frac{50 (25 x^{2} - 4) x}{| 25 x^{2} - 4 |}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{(50 (25 x^{2}) + 50 \cdot - 4) x}{| 25 x^{2} - 4 |}$

Step 1.1.3.2

Apply the distributive property.

$\frac{50 (25 x^{2}) x + 50 \cdot - 4 x}{| 25 x^{2} - 4 |}$

Step 1.1.3.3

Simplify each term.

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Step 1.1.3.3.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.3.3.1.1

Move $x$ .

$\frac{50 (25 (x \cdot x^{2})) + 50 \cdot - 4 x}{| 25 x^{2} - 4 |}$

Step 1.1.3.3.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.3.3.1.2.1

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

$\frac{50 (25 (x^{1} x^{2})) + 50 \cdot - 4 x}{| 25 x^{2} - 4 |}$

Step 1.1.3.3.1.2.2

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

$\frac{50 (25 x^{1 + 2}) + 50 \cdot - 4 x}{| 25 x^{2} - 4 |}$

Step 1.1.3.3.1.3

Add $1$ and $2$ .

$\frac{50 (25 x^{3}) + 50 \cdot - 4 x}{| 25 x^{2} - 4 |}$

Step 1.1.3.3.2

Multiply $25$ by $50$ .

$\frac{1250 x^{3} + 50 \cdot - 4 x}{| 25 x^{2} - 4 |}$

Step 1.1.3.3.3

Multiply $50$ by $- 4$ .

$f' (x) = \frac{1250 x^{3} - 200 x}{| 25 x^{2} - 4 |}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1250 x^{3} - 200 x}{| 25 x^{2} - 4 |}$ .

$\frac{1250 x^{3} - 200 x}{| 25 x^{2} - 4 |}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{1250 x^{3} - 200 x}{| 25 x^{2} - 4 |} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{1250 x^{3} - 200 x}{| 25 x^{2} - 4 |} = 0$

Step 2.2

Set the numerator equal to zero.

$1250 x^{3} - 200 x = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Factor the left side of the equation.

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Step 2.3.1.1

Factor $50 x$ out of $1250 x^{3} - 200 x$ .

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Step 2.3.1.1.1

Factor $50 x$ out of $1250 x^{3}$ .

$50 x (25 x^{2}) - 200 x = 0$

Step 2.3.1.1.2

Factor $50 x$ out of $- 200 x$ .

$50 x (25 x^{2}) + 50 x (- 4) = 0$

Step 2.3.1.1.3

Factor $50 x$ out of $50 x (25 x^{2}) + 50 x (- 4)$ .

$50 x (25 x^{2} - 4) = 0$

Step 2.3.1.2

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$50 x ({(5 x)}^{2} - 4) = 0$

Step 2.3.1.3

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

$50 x ({(5 x)}^{2} - 2^{2}) = 0$

Step 2.3.1.4

Factor.

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Step 2.3.1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x$ and $b = 2$ .

$50 x ((5 x + 2) (5 x - 2)) = 0$

Step 2.3.1.4.2

Remove unnecessary parentheses.

$50 x (5 x + 2) (5 x - 2) = 0$

Step 2.3.2

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

$x = 0$

$5 x + 2 = 0$

$5 x - 2 = 0$

Step 2.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.4

Set $5 x + 2$ equal to $0$ and solve for $x$ .

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Step 2.3.4.1

Set $5 x + 2$ equal to $0$ .

$5 x + 2 = 0$

Step 2.3.4.2

Solve $5 x + 2 = 0$ for $x$ .

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Step 2.3.4.2.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 2.3.4.2.2

Divide each term in $5 x = - 2$ by $5$ and simplify.

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Step 2.3.4.2.2.1

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 2.3.4.2.2.2

Simplify the left side.

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Step 2.3.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.3.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 2.3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Step 2.3.4.2.2.3

Simplify the right side.

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Step 2.3.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 2.3.5

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

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Step 2.3.5.1

Set $5 x - 2$ equal to $0$ .

$5 x - 2 = 0$

Step 2.3.5.2

Solve $5 x - 2 = 0$ for $x$ .

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Step 2.3.5.2.1

Add $2$ to both sides of the equation.

$5 x = 2$

Step 2.3.5.2.2

Divide each term in $5 x = 2$ by $5$ and simplify.

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Step 2.3.5.2.2.1

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Step 2.3.5.2.2.2

Simplify the left side.

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Step 2.3.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Step 2.3.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 2.3.6

The final solution is all the values that make $50 x (5 x + 2) (5 x - 2) = 0$ true.

$x = 0, - \frac{2}{5}, \frac{2}{5}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{1250 x^{3} - 200 x}{| 25 x^{2} - 4 |}$ equal to $0$ to find where the expression is undefined.

$| 25 x^{2} - 4 | = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$25 x^{2} - 4 = \pm 0$

Step 3.2.2

Plus or minus $0$ is $0$ .

$25 x^{2} - 4 = 0$

Step 3.2.3

Add $4$ to both sides of the equation.

$25 x^{2} = 4$

Step 3.2.4

Divide each term in $25 x^{2} = 4$ by $25$ and simplify.

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Step 3.2.4.1

Divide each term in $25 x^{2} = 4$ by $25$ .

$\frac{25 x^{2}}{25} = \frac{4}{25}$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $25$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

$\frac{25 x^{2}}{25} = \frac{4}{25}$

Step 3.2.4.2.1.2

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

$x^{2} = \frac{4}{25}$

Step 3.2.5

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

$x = \pm \sqrt{\frac{4}{25}}$

Step 3.2.6

Simplify $\pm \sqrt{\frac{4}{25}}$ .

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Step 3.2.6.1

Rewrite $\sqrt{\frac{4}{25}}$ as $\frac{\sqrt{4}}{\sqrt{25}}$ .

$x = \pm \frac{\sqrt{4}}{\sqrt{25}}$

Step 3.2.6.2

Simplify the numerator.

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Step 3.2.6.2.1

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

$x = \pm \frac{\sqrt{2^{2}}}{\sqrt{25}}$

Step 3.2.6.2.2

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

$x = \pm \frac{2}{\sqrt{25}}$

Step 3.2.6.3

Simplify the denominator.

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Step 3.2.6.3.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \frac{2}{\sqrt{5^{2}}}$

Step 3.2.6.3.2

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

$x = \pm \frac{2}{5}$

Step 3.2.7

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

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Step 3.2.7.1

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

$x = \frac{2}{5}$

Step 3.2.7.2

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

$x = - \frac{2}{5}$

Step 3.2.7.3

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

$x = \frac{2}{5}, - \frac{2}{5}$

Step 3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - \frac{2}{5}, x = \frac{2}{5}$

Step 4

Evaluate $| 25 x^{2} - 4 |$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$| 25 {(0)}^{2} - 4 |$

Step 4.1.2

Simplify.

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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 $0$ .

$| 25 \cdot 0 - 4 |$

Step 4.1.2.1.2

Multiply $25$ by $0$ .

$| 0 - 4 |$

Step 4.1.2.2

Subtract $4$ from $0$ .

$| - 4 |$

Step 4.1.2.3

The absolute value is the distance between a number and zero. The distance between $- 4$ and $0$ is $4$ .

$4$

Step 4.2

Evaluate at $x = - \frac{2}{5}$ .

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Step 4.2.1

Substitute $- \frac{2}{5}$ for $x$ .

$| 25 {(- \frac{2}{5})}^{2} - 4 |$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

Apply the product rule to $- \frac{2}{5}$ .

$| 25 ({(- 1)}^{2} {(\frac{2}{5})}^{2}) - 4 |$

Step 4.2.2.1.1.2

Apply the product rule to $\frac{2}{5}$ .

$| 25 ({(- 1)}^{2} \frac{2^{2}}{5^{2}}) - 4 |$

Step 4.2.2.1.2

Raise $- 1$ to the power of $2$ .

$| 25 (1 \frac{2^{2}}{5^{2}}) - 4 |$

Step 4.2.2.1.3

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

$| 25 \frac{2^{2}}{5^{2}} - 4 |$

Step 4.2.2.1.4

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

$| 25 \frac{4}{5^{2}} - 4 |$

Step 4.2.2.1.5

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

$| 25 (\frac{4}{25}) - 4 |$

Step 4.2.2.1.6

Cancel the common factor of $25$ .

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Step 4.2.2.1.6.1

Cancel the common factor.

$| 25 (\frac{4}{25}) - 4 |$

Step 4.2.2.1.6.2

Rewrite the expression.

$| 4 - 4 |$

Step 4.2.2.2

Subtract $4$ from $4$ .

$| 0 |$

Step 4.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$0$

Step 4.3

Evaluate at $x = \frac{2}{5}$ .

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Step 4.3.1

Substitute $\frac{2}{5}$ for $x$ .

$| 25 {(\frac{2}{5})}^{2} - 4 |$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Apply the product rule to $\frac{2}{5}$ .

$| 25 \frac{2^{2}}{5^{2}} - 4 |$

Step 4.3.2.1.2

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

$| 25 \frac{4}{5^{2}} - 4 |$

Step 4.3.2.1.3

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

$| 25 (\frac{4}{25}) - 4 |$

Step 4.3.2.1.4

Cancel the common factor of $25$ .

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Step 4.3.2.1.4.1

Cancel the common factor.

$| 25 (\frac{4}{25}) - 4 |$

Step 4.3.2.1.4.2

Rewrite the expression.

$| 4 - 4 |$

Step 4.3.2.2

Subtract $4$ from $4$ .

$| 0 |$

Step 4.3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$0$

Step 4.4

List all of the points.

$(0, 4), (- \frac{2}{5}, 0), (\frac{2}{5}, 0)$

Step 5