Calculus Examples

$f (x) = \frac{32}{x^{2} - 6 x - 7}$

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 Constant Multiple Rule.

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Step 1.1.1.1

Since $32$ is constant with respect to $x$ , the derivative of $\frac{32}{x^{2} - 6 x - 7}$ with respect to $x$ is $32 \frac{d}{d x} [\frac{1}{x^{2} - 6 x - 7}]$ .

$32 \frac{d}{d x} [\frac{1}{x^{2} - 6 x - 7}]$

Step 1.1.1.2

Rewrite $\frac{1}{x^{2} - 6 x - 7}$ as ${(x^{2} - 6 x - 7)}^{- 1}$ .

$32 \frac{d}{d x} [{(x^{2} - 6 x - 7)}^{- 1}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} - 6 x - 7$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 6 x - 7$ .

$32 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 6 x - 7])$

Step 1.1.2.2

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

$32 (- u^{- 2} \frac{d}{d x} [x^{2} - 6 x - 7])$

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{2} - 6 x - 7$ .

$32 (- {(x^{2} - 6 x - 7)}^{- 2} \frac{d}{d x} [x^{2} - 6 x - 7])$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Multiply $- 1$ by $32$ .

$- 32 ({(x^{2} - 6 x - 7)}^{- 2} \frac{d}{d x} [x^{2} - 6 x - 7])$

Step 1.1.3.2

By the Sum Rule, the derivative of $x^{2} - 6 x - 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 7]$ .

$- 32 {(x^{2} - 6 x - 7)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 7])$

Step 1.1.3.3

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

$- 32 {(x^{2} - 6 x - 7)}^{- 2} (2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 7])$

Step 1.1.3.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$- 32 {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])$

Step 1.1.3.5

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

$- 32 {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6 \cdot 1 + \frac{d}{d x} [- 7])$

Step 1.1.3.6

Multiply $- 6$ by $1$ .

$- 32 {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6 + \frac{d}{d x} [- 7])$

Step 1.1.3.7

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$- 32 {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6 + 0)$

Step 1.1.3.8

Add $2 x - 6$ and $0$ .

$- 32 {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6)$

Step 1.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 32 \frac{1}{{(x^{2} - 6 x - 7)}^{2}} (2 x - 6)$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Combine terms.

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Step 1.1.5.1.1

Combine $- 32$ and $\frac{1}{{(x^{2} - 6 x - 7)}^{2}}$ .

$\frac{- 32}{{(x^{2} - 6 x - 7)}^{2}} (2 x - 6)$

Step 1.1.5.1.2

Move the negative in front of the fraction.

$- \frac{32}{{(x^{2} - 6 x - 7)}^{2}} (2 x - 6)$

Step 1.1.5.2

Reorder the factors of $- \frac{32}{{(x^{2} - 6 x - 7)}^{2}} (2 x - 6)$ .

$f' (x) = - (2 x - 6) \frac{32}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- (2 x - 6) \frac{32}{{(x^{2} - 6 x - 7)}^{2}}$ .

$- (2 x - 6) \frac{32}{{(x^{2} - 6 x - 7)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- (2 x - 6) \frac{32}{{(x^{2} - 6 x - 7)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- (2 x - 6) \frac{32}{{(x^{2} - 6 x - 7)}^{2}} = 0$

Step 2.2

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

$2 x - 6 = 0$

$\frac{32}{{(x^{2} - 6 x - 7)}^{2}} = 0$

Step 2.3

Set $2 x - 6$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $2 x - 6$ equal to $0$ .

$2 x - 6 = 0$

Step 2.3.2

Solve $2 x - 6 = 0$ for $x$ .

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Step 2.3.2.1

Add $6$ to both sides of the equation.

$2 x = 6$

Step 2.3.2.2

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 2.3.2.2.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Divide $6$ by $2$ .

$x = 3$

Step 2.4

Set $\frac{32}{{(x^{2} - 6 x - 7)}^{2}}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $\frac{32}{{(x^{2} - 6 x - 7)}^{2}}$ equal to $0$ .

$\frac{32}{{(x^{2} - 6 x - 7)}^{2}} = 0$

Step 2.4.2

Solve $\frac{32}{{(x^{2} - 6 x - 7)}^{2}} = 0$ for $x$ .

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Step 2.4.2.1

Set the numerator equal to zero.

$32 = 0$

Step 2.4.2.2

Since $32 \neq 0$ , there are no solutions.

No solution

Step 2.5

The final solution is all the values that make $- (2 x - 6) \frac{32}{{(x^{2} - 6 x - 7)}^{2}} = 0$ true.

$x = 3$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{32}{{(x^{2} - 6 x - 7)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{2} - 6 x - 7)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Factor the left side of the equation.

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Step 3.2.1.1

Factor $x^{2} - 6 x - 7$ using the AC method.

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Step 3.2.1.1.1

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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 3.2.1.1.2

Write the factored form using these integers.

${((x - 7) (x + 1))}^{2} = 0$

Step 3.2.1.2

Apply the product rule to $(x - 7) (x + 1)$ .

${(x - 7)}^{2} {(x + 1)}^{2} = 0$

Step 3.2.2

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

${(x - 7)}^{2} = 0$

${(x + 1)}^{2} = 0$

Step 3.2.3

Set ${(x - 7)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set ${(x - 7)}^{2}$ equal to $0$ .

${(x - 7)}^{2} = 0$

Step 3.2.3.2

Solve ${(x - 7)}^{2} = 0$ for $x$ .

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Step 3.2.3.2.1

Set the $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 3.2.3.2.2

Add $7$ to both sides of the equation.

$x = 7$

Step 3.2.4

Set ${(x + 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.4.1

Set ${(x + 1)}^{2}$ equal to $0$ .

${(x + 1)}^{2} = 0$

Step 3.2.4.2

Solve ${(x + 1)}^{2} = 0$ for $x$ .

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Step 3.2.4.2.1

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 3.2.4.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.2.5

The final solution is all the values that make ${(x - 7)}^{2} {(x + 1)}^{2} = 0$ true.

$x = 7, - 1$

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 $0$ .

$x = - 1, x = 7$

Step 4

Evaluate $\frac{32}{x^{2} - 6 x - 7}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 3$ .

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Step 4.1.1

Substitute $3$ for $x$ .

$\frac{32}{{(3)}^{2} - 6 \cdot 3 - 7}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the denominator.

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Step 4.1.2.1.1

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

$\frac{32}{9 - 6 \cdot 3 - 7}$

Step 4.1.2.1.2

Multiply $- 6$ by $3$ .

$\frac{32}{9 - 18 - 7}$

Step 4.1.2.1.3

Subtract $18$ from $9$ .

$\frac{32}{- 9 - 7}$

Step 4.1.2.1.4

Subtract $7$ from $- 9$ .

$\frac{32}{- 16}$

Step 4.1.2.2

Divide $32$ by $- 16$ .

$- 2$

Step 4.2

Evaluate at $x = - 1$ .

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Step 4.2.1

Substitute $- 1$ for $x$ .

$\frac{32}{{(- 1)}^{2} - 6 \cdot - 1 - 7}$

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

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

$\frac{32}{1 - 6 \cdot - 1 - 7}$

Step 4.2.2.1.2

Multiply $- 6$ by $- 1$ .

$\frac{32}{1 + 6 - 7}$

Step 4.2.2.2

Simplify by adding and subtracting.

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Step 4.2.2.2.1

Add $1$ and $6$ .

$\frac{32}{7 - 7}$

Step 4.2.2.2.2

Subtract $7$ from $7$ .

$\frac{32}{0}$

Step 4.2.2.2.3

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

Undefined

$\frac{32}{0}$

Step 4.2.2.3

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

Undefined

Step 4.3

Evaluate at $x = 7$ .

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Step 4.3.1

Substitute $7$ for $x$ .

$\frac{32}{{(7)}^{2} - 6 \cdot 7 - 7}$

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

Raise $7$ to the power of $2$ .

$\frac{32}{49 - 6 \cdot 7 - 7}$

Step 4.3.2.1.2

Multiply $- 6$ by $7$ .

$\frac{32}{49 - 42 - 7}$

Step 4.3.2.2

Simplify by subtracting numbers.

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Step 4.3.2.2.1

Subtract $42$ from $49$ .

$\frac{32}{7 - 7}$

Step 4.3.2.2.2

Subtract $7$ from $7$ .

$\frac{32}{0}$

Step 4.3.2.2.3

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

Undefined

$\frac{32}{0}$

Step 4.3.2.3

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

Undefined

Step 4.4

List all of the points.

$(3, - 2)$

Step 5