Calculus Examples

$f (x) = \frac{x^{2} - 9}{x - 5}$

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 $f (x) = x^{2} - 9$ and $g (x) = x - 5$ .

$\frac{(x - 5) \frac{d}{d x} [x^{2} - 9] - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\frac{(x - 5) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]) - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

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 $n = 2$ .

$\frac{(x - 5) (2 x + \frac{d}{d x} [- 9]) - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 1.1.2.3

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

$\frac{(x - 5) (2 x + 0) - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $2 x$ and $0$ .

$\frac{(x - 5) (2 x) - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 1.1.2.4.2

Move $2$ to the left of $x - 5$ .

$\frac{2 \cdot (x - 5) x - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 1.1.2.5

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

$\frac{2 (x - 5) x - (x^{2} - 9) (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 1.1.2.6

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

$\frac{2 (x - 5) x - (x^{2} - 9) (1 + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 1.1.2.7

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{2 (x - 5) x - (x^{2} - 9) (1 + 0)}{{(x - 5)}^{2}}$

Step 1.1.2.8

Simplify the expression.

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Step 1.1.2.8.1

Add $1$ and $0$ .

$\frac{2 (x - 5) x - (x^{2} - 9) \cdot 1}{{(x - 5)}^{2}}$

Step 1.1.2.8.2

Multiply $- 1$ by $1$ .

$\frac{2 (x - 5) x - (x^{2} - 9)}{{(x - 5)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot - 5) x - (x^{2} - 9)}{{(x - 5)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$\frac{2 x \cdot x + 2 \cdot - 5 x - (x^{2} - 9)}{{(x - 5)}^{2}}$

Step 1.1.3.3

Apply the distributive property.

$\frac{2 x \cdot x + 2 \cdot - 5 x - x^{2} - - 9}{{(x - 5)}^{2}}$

Step 1.1.3.4

Simplify the numerator.

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Step 1.1.3.4.1

Simplify each term.

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Step 1.1.3.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.3.4.1.1.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 \cdot - 5 x - x^{2} - - 9}{{(x - 5)}^{2}}$

Step 1.1.3.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 \cdot - 5 x - x^{2} - - 9}{{(x - 5)}^{2}}$

Step 1.1.3.4.1.2

Multiply $2$ by $- 5$ .

$\frac{2 x^{2} - 10 x - x^{2} - - 9}{{(x - 5)}^{2}}$

Step 1.1.3.4.1.3

Multiply $- 1$ by $- 9$ .

$\frac{2 x^{2} - 10 x - x^{2} + 9}{{(x - 5)}^{2}}$

Step 1.1.3.4.2

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

$\frac{x^{2} - 10 x + 9}{{(x - 5)}^{2}}$

Step 1.1.3.5

Factor $x^{2} - 10 x + 9$ using the AC method.

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Step 1.1.3.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 1.1.3.5.2

Write the factored form using these integers.

$f' (x) = \frac{(x - 9) (x - 1)}{{(x - 5)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{(x - 9) (x - 1)}{{(x - 5)}^{2}}$ .

$\frac{(x - 9) (x - 1)}{{(x - 5)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{(x - 9) (x - 1)}{{(x - 5)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{(x - 9) (x - 1)}{{(x - 5)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$(x - 9) (x - 1) = 0$

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 $0$ .

$x - 9 = 0$

$x - 1 = 0$

Step 2.3.2

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 2.3.2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 2.3.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.3.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.3.4

The final solution is all the values that make $(x - 9) (x - 1) = 0$ true.

$x = 9, 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{(x - 9) (x - 1)}{{(x - 5)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$x - 5 = 0$

Step 3.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4

Evaluate $\frac{x^{2} - 9}{x - 5}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 9$ .

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Step 4.1.1

Substitute $9$ for $x$ .

$\frac{{(9)}^{2} - 9}{(9) - 5}$

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 $9$ to the power of $2$ .

$\frac{81 - 9}{9 - 5}$

Step 4.1.2.1.2

Subtract $9$ from $81$ .

$\frac{72}{9 - 5}$

Step 4.1.2.2

Simplify the expression.

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Step 4.1.2.2.1

Subtract $5$ from $9$ .

$\frac{72}{4}$

Step 4.1.2.2.2

Divide $72$ by $4$ .

$18$

Step 4.2

Evaluate at $x = 1$ .

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Step 4.2.1

Substitute $1$ for $x$ .

$\frac{{(1)}^{2} - 9}{(1) - 5}$

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

One to any power is one.

$\frac{1 - 9}{1 - 5}$

Step 4.2.2.1.2

Subtract $9$ from $1$ .

$\frac{- 8}{1 - 5}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Subtract $5$ from $1$ .

$\frac{- 8}{- 4}$

Step 4.2.2.2.2

Divide $- 8$ by $- 4$ .

$2$

Step 4.3

Evaluate at $x = 5$ .

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Step 4.3.1

Substitute $5$ for $x$ .

$\frac{{(5)}^{2} - 9}{(5) - 5}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Subtract $5$ from $5$ .

$\frac{{(5)}^{2} - 9}{0}$

Step 4.3.2.2

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

Undefined

Step 4.4

List all of the points.

$(9, 18), (1, 2)$

Step 5