Calculus Examples

$f (x) = \frac{x^{2} + 25}{x}$

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 Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} + 25$ and $g (x) = x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

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 $n = 2$ .

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

Step 1.1.2.3

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$\frac{x (2 x + 0) - (x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.4

Add $2 x$ and $0$ .

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

Step 1.1.3

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

$\frac{2 (x^{1} x) - (x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.4

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

$\frac{2 (x^{1} x^{1}) - (x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.5

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

$\frac{2 x^{1 + 1} - (x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.6

Add $1$ and $1$ .

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

Step 1.1.7

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

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

Step 1.1.8

Multiply $- 1$ by $1$ .

$\frac{2 x^{2} - (x^{2} + 25)}{x^{2}}$

Step 1.1.9

Simplify.

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Step 1.1.9.1

Apply the distributive property.

$\frac{2 x^{2} - x^{2} - 1 \cdot 25}{x^{2}}$

Step 1.1.9.2

Simplify the numerator.

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Step 1.1.9.2.1

Multiply $- 1$ by $25$ .

$\frac{2 x^{2} - x^{2} - 25}{x^{2}}$

Step 1.1.9.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} - 25}{x^{2}}$

Step 1.1.9.3

Simplify the numerator.

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Step 1.1.9.3.1

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

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

Step 1.1.9.3.2

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

$f' (x) = \frac{(x + 5) (x - 5)}{x^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{(x + 5) (x - 5)}{x^{2}}$ .

$\frac{(x + 5) (x - 5)}{x^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{(x + 5) (x - 5)}{x^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

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

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

$x + 5 = 0$

$x - 5 = 0$

Step 2.3.2

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

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Step 2.3.2.1

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

$x + 5 = 0$

Step 2.3.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.3.3

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

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Step 2.3.3.1

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

$x - 5 = 0$

Step 2.3.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.3.4

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

$x = - 5, 5$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{(x + 5) (x - 5)}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $\frac{x^{2} + 25}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 5$ .

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Step 4.1.1

Substitute $- 5$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

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

$\frac{25 + 25}{- 5}$

Step 4.1.2.1.2

Add $25$ and $25$ .

$\frac{50}{- 5}$

Step 4.1.2.2

Divide $50$ by $- 5$ .

$- 10$

Step 4.2

Evaluate at $x = 5$ .

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Step 4.2.1

Substitute $5$ for $x$ .

$\frac{{(5)}^{2} + 25}{5}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

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

$\frac{25 + 25}{5}$

Step 4.2.2.1.2

Add $25$ and $25$ .

$\frac{50}{5}$

Step 4.2.2.2

Divide $50$ by $5$ .

$10$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

$\frac{{(0)}^{2} + 25}{0}$

Step 4.3.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.4

List all of the points.

$(- 5, - 10), (5, 10)$

Step 5